Unimodular Hausdorff and Minkowski dimensions

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Electronic
J
ournal
of
Probability
Electron. J. Probab. 26 (2021), article no. 155, 1–64.
ISSN: 1083-6489 https://doi.org/10.1214/21-EJP692
Unimodular Hausdorff and Minkowski dimensions*
François Baccelli†Mir-Omid Haji-Mirsadeghi‡Ali Khezeli§ ¶
Abstract
This work introduces two new notions of dimension, namely the unimodular Minkowski
and Hausdorff dimensions, which are inspired from the classical analogous notions.
These dimensions are defined for unimodular discrete spaces, introduced in this work,
which provide a common generalization to stationary point processes under their
Palm version and unimodular random rooted graphs. The use of unimodularity in the
definitions of dimension is novel. Also, a toolbox of results is presented for the analysis
of these dimensions. In particular, analogues of Billingsley’s lemma and Frostman’s
lemma are presented. These last lemmas are instrumental in deriving upper bounds
on dimensions, whereas lower bounds are obtained from specific coverings. The
notions of unimodular Hausdorff size, which is a discrete analogue of the Hausdorff
measure, and unimodular dimension function are also introduced. This toolbox allows
one to connect the unimodular dimensions to other notions such as volume growth
rate, discrete dimension and scaling limits. It is also used to analyze the dimensions
of a set of examples pertaining to point processes, branching processes, random
graphs, random walks, and self-similar discrete random spaces. Further results of
independent interest are also presented, like a version of the max-flow min-cut theorem
for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all
unimodular discrete spaces.
Keywords:
random discrete metric space; mass transport principle; infinite random graph; point
stationary point process; Palm calculus; self-similar sets; random walks.
MSC2020 subject classifications: 60D05; 05C63; 28A78.
Submitted to EJP on February 8, 2020, final version accepted on August 27, 2021.
Supersedes arXiv:https://arxiv.org/abs/1807.02980.
*
Supported in part by a grant of the Simons Foundation (#197982) to The University of Texas at Austin and
by the ERC NEMO grant, under the European Union’s Horizon 2020 research and innovation programme, grant
agreement number 788851 to INRIA. The second author thanks the Research and Technology Vice-presidency
of Sharif University of Technology for its support. This research was done while the third author was affiliatd
with Tarbiat Modares University and was in part supported by a grant from IPM (No.98490118). He is currently
affiliated with INRIA Paris.
†INRIA Paris and the University of Texas at Austin. E-mail: francois.baccelli@ens.fr
‡Sharif University of Technology. E-mail: mirsadeghi@sharif.ir
§Tarbiat Modares University. E-mail: khezeli@modares.ac.ir
¶Current address: INRIA Paris. E-mail: ali.khezeli@inria.fr

Unimodular Hausdorff and Minkowski dimensions
1 Introduction
Infinite discrete random structures are ubiquitous: random graphs, branching pro-
cesses, point processes, graphs or zeros of discrete random walks, discrete or continuum
percolation, to name a few. The large scale and macroscopic properties of such spaces
have been thoroughly discussed in the literature. In particular, various notions of di-
mension have been proposed; e.g., the mass dimension and the discrete (Hausdorff)
dimension defined by Barlow and Taylor [8] for subsets of Zd.
The main novelty of the present paper is the definition of new notions of dimension
for a class of discrete structures that, heuristically, enjoy a form of statistical homo-
geneity. The mathematical framework proposed to handle such structures is that of
unimodular (random) discrete spaces, where unimodularity is defined here by a version
of the mass transport principle. This framework unifies unimodular random graphs and
networks, stationary point processes (under their Palm version) and point-stationary
point processes. It does not require more than a metric; for instance, no edges or no
underlying Euclidean spaces are needed. The statistical homogeneity of such spaces has
been used to define localized versions of global notions such as intensity. The novelty
of the present paper is the use of this homogeneity to define the notions of unimodular
Minkowski and Hausdorff dimensions, which are inspired by the analogous classical
notions. The definitions are obtained naturally from the classical setting by replacing
the infinite sums pertaining to infinite coverings by the expectation of certain random
variables at the origin (which is a distinguished point), and also by considering large
balls instead of small balls. These definitions are local but capture macroscopic (large
scale) properties of the space.
The definitions are complemented by a toolbox for the analysis of unimodular dimen-
sions. Several analogues of the important results known about the classical Hausdorff
and Minkowski dimensions are established, like for instance the comparison of the
unimodular Minkowski and Hausdorff dimensions as well as unimodular versions of
Billingsley’s lemma and Frostman’s lemma. These lemmas allow one to connect the
dimension to the (polynomial) volume growth rate of the space, which is also called
mass dimension or fractal dimension in the literature. While many ideas in this toolbox
are imported from the continuum setting, their adaptation is nontrivial and there is
no automatic way to import results from the continuum to the discrete setting. For
some results, the statements fundamentally differ from their continuum analog; e.g., the
statement of Billingsley’s lemma.
These notions of dimension are complemented by further definitions which can be
used for a finer study of dimension. An analogue of the Hausdorff measure is defined,
which is called the unimodular Hausdorff size here. This can be used to compare
sets with the same dimension. The notion of unimodular dimension function is also
defined for a finer quantification of the dimension. Such notions are new for discrete
spaces to the best of the authors’ knowledge. Another new notion introduced in the
present paper is that of regularity for unimodular spaces, which is the equality of the
unimodular Minkowski and Hausdorff dimensions. Similar notions of regularity exist in
the continuum setting (see e.g., the definition of fractals in [
14
]) and for subsets of
Zd
[9].
The paper also contains new mathematical results of independent interest. A weak
version of Birkhoff’s pointwise ergodic theorem is stated for all unimodular discrete
spaces. A unimodular version of the max-flow min-cut theorem is also proved for
unimodular one-ended trees, which is used in the proof of the unimodular Frostman
lemma. Also, for unimodular one-ended trees, a relation between the volume growth
rate and the height of the root is established as explained below.
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Unimodular Hausdorff and Minkowski dimensions
The framework is used to derive concrete results on the dimension of several in-
stances of unimodular random discrete metric spaces. This is done for the zeros and
the graph of discrete random walks, sets defined by digit restriction, trees obtained
from branching processes and drainage network models, etc. Some general results are
obtained for all unimodular trees. For instance, a general relation is established between
the unimodular dimensions of a unimodular one-ended tree and the tail of the distri-
bution of the height of the root. The dimensions of some unimodular discrete random
self-similar sets are also discussed. The latter are defined in this paper as unimodular
discrete analogues of self similar sets such as the Koch snowflake, the Sierpinski triangle,
etc.
This framework opens several further research directions. Firstly, it might be useful
for the study of some discrete examples which are of interest in mathematical physics.
Many examples in this domain enjoy some kind of homogeneity and give rise to unimod-
ular spaces, directly or indirectly; e.g., percolation clusters and self-avoiding random
walks. A few such examples are studied in detail in this work and in the preprint [
6
].
One might expect that in these examples, the values of unimodular dimensions match
the conjectures or results pertaining to other notions of dimension that are applicable.
Secondly, the definitions and many of the results are valid for exponential (or other)
gauge functions. The proposed framework might hence have applications in group theory
(or other areas), where most interesting examples have super-polynomial growth. A third
important line of thoughts is the connections of unimodular dimensions to other notions
of dimension. Some first connections are discussed in Subsection 8.1. The preprint [
6
]
discusses ongoing research on these questions as well as further developments of these
notions of dimensions.
1.1 Summary of the Main Definitions and Results
Recall that the ordinary Minkowski dimension of a compact metric space
X
is defined
using the minimum number of balls of radius

needed to cover
X
. Now, consider a
(unimodular) discrete space
D
(it is useful to have in mind the example
D=Zk
to see
how the definitions work). It is convenient to consider coverings of
D
by balls of equal
but large radius. Of course, if
D
is unbounded, then an infinite number of balls is needed
to cover
D
. So one needs another measure to assess how many balls are used in a
covering. Let
S⊆D
be the set of centers of the balls in the covering. The idea pursued
in this paper is that if
D
is unimodular, then the intensity of
S
is a measure of the
average number of points of
S
per points of
D
(
S
should be equivariant for the intensity
to be defined, as discussed later). This gives rise to the definition of the unimodular
Minkowski dimension naturally.
The idea behind the definition of the unimodular Hausdorff dimension is similar.
Recall that the
α
-dimensional Hausdorff content of a compact metric space
X
is defined
by considering the infimum of
PiRα
i
, where the
Ri
’s are the radii of a sequence of balls
that cover
X
. Also, it is convenient to enforce an upper bound on the radii. Now, consider
a unimodular discrete space
D
and a covering of
D
by balls which may have different
radii. Let
R(v)
be the radius of the ball centered at
v
. It is convenient to consider a lower
bound on the radii, say
R(·)≥1
. Again, if
D
is unbounded, then
PvR(v)α
is always
infinite. The idea is to leverage the unimodularity of
D
and to consider the average
of the values
R(·)α
per point as a replacement of the sum. Under the unimodularity
assumption, this can be defined by
E[R(o)α]
, where
o
stands for the distinguished point
of
D
(called the origin) and where, by convention,
R(o)
is zero if there is no ball centered
at o. This is used to define the unimodular Hausdorff dimension of Din a natural way.
The volume growth rate of the space is the polynomial growth rate of
#Nr(o)
,
where
Nr(o)
represents the closed ball of radius
r
centered at the origin and
#Nr(o)
EJP 26 (2021), paper 155. Page 3/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
is the number of points in this ball. It is shown that the upper and lower volume
growth rates of
#Nr(o)
(i.e., limsup and liminf of
log(#Nr(o))/log r
as
r→ ∞
) provide
upper and lower bound for the unimodular Hausdorff dimension, respectively. This
is a discrete analogue of Billingsley’s lemma (see e.g., [
14
]). A discrete analogue
of the mass distribution principle is also provided, which is useful to derive upper
bounds on the unimodular Hausdorff dimension. In the Euclidean case (i.e., for point-
stationary point processes equipped with the Euclidean metric), it is shown that the
unimodular Minkowski dimension is bounded from above by the polynomial decay rate
of
E[1/#Nn(o)]
. Weighted versions of these inequalities, where a weight is assigned
to each point, are also presented. As a corollary, a weak form of Birkhoff’s pointwise
ergodic theorem is established for all unimodular discrete spaces. These results are very
useful for calculating the unimodular dimensions in many examples. An important result
is an analogue of Frostman’s lemma. Roughly speaking, this lemma states that the mass
distribution principle is sharp if the weights are chosen appropriately. This lemma is
a powerful tool to study the unimodular Hausdorff dimension. In the Euclidean case,
another proof of Frostman’s lemma is provided using a version of the max-flow min-cut
theorem for unimodular one-ended trees, which is of independent interest.
Depending on whether one defines the unimodular Minkowski dimension as the decay
rate or the growth rate of the optimal intensity of the coverings by balls of radius
r
,
one gets positive or negative dimensions. The present paper adopts the convention of
positive dimensions for the definitions of both the unimodular Minkowski and Hausdorff
dimensions, despite some mathematical arguments in favor of negative dimensions.
Further discussion on the matter is provided in Subsection 8.3.
1.2 Organization of the Material
Section 2 defines unimodular discrete spaces and equivariant processes, which are
needed throughout. Section 3 presents the definitions of the unimodular Minkowski
and Hausdorff dimensions and the unimodular Hausdorff size. It also provides some
basic properties of these unimodular dimensions as part of the toolbox for the analysis
of unimodular dimensions. Various examples are discussed in Section 4. These examples
are used throughout the paper. Section 5 is focused on the connections with volume
growth rates and contains the statements and proofs of the unimodular Billingsey lemma
and of the mass distribution principle. The unimodulat Frostman lemma is discussed in
Section 7. Section 6 completes the analysis of the examples discussed in Section 4 and
also discusses new examples for further illustration of the results. Section 8 discusses
further topics on the matter. This includes a discussion of the connections to earlier
notions of dimensions for discrete sets, in particular those proposed by Barlow and Taylor
in [
8
,
9
], as well as a discussion on negative dimensions. A collection of conjectures and
open problems are also listed in this section.
Throughout the paper, some easier proofs as well as some extra details (e.g., measure-
theoretic requirements) are skipped for the sake of brevity. These proofs and details are
nevertheless available in the arXiv version [
7
] of the present paper. Precise indications
on where to find them in these preprints are given in the text.
2 Unimodular Discrete Spaces
The main objective of this section is the definition of unimodular discrete spaces as a
common generalization of unimodular graphs, Palm probabilities and point-stationary
point processes. If the reader is familiar with unimodular random graphs, he or she can
restrict attention to the case of unimodular graphs and jump to Subsection 2.5 at first
reading.
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Unimodular Hausdorff and Minkowski dimensions
2.1 Notation and Definitions
The following notation will be used throughout. The set of nonnegative real (resp.
integer) numbers is denoted by
R≥0
(resp.
Z≥0
). The minimum and maximum binary
operators are denoted by
∧
and
∨
respectively. The number of elements in a set
A
is
denoted by
#A
, which is a number in
[0,∞]
. If
P(x)
is a property about
x
, the indicator
1{P(x)}is equal to 1 if P(x)is true and 0 otherwise.
Discrete metric spaces (discussed in details in Subsection 2.2) are denoted by
D
,
D0
, etc. Graphs are an important class of discrete metric spaces. So the symbols and
notations are mostly borrowed from graph theory.
For
r > 0
,
Nr(v) :=Nr(D, v)
denotes the closed
r
-neighborhood of
v∈D
; i.e., the set
of points of
D
with distance less than or equal to
r
from
v
. An exception is made for
r= 0
(Subsection 3.3), where
N0(v) := ∅
. The diameter of a subset
A
is denoted by
diam(A)
.
For a function
f: [1,∞)→R≥0
, the polynomial growth rates and polynomial decay rates
are defined by the following formulas:
growth (f) := −decay (f) := liminf
r→∞ log f(r)/log r,
growth (f) := −decay (f) := limsup
r→∞
log f(r)/log r,
growth (f) := −decay (f) := lim
r→∞ log f(r)/log r.
Definition 2.1.
Let
µ
be a probability measure on a measurable space
X
and
w:X→
R≥0
be a measurable function. Assume
0< c := RXw(x)dµ(x)<∞
. By
biasing µby w
we mean the probability measure νon Xdefined by
ν(A) := 1
cZA
w(x)dµ(x).
2.2 The Space of Pointed Discrete Spaces
Throughout the paper, the metric on any metric space is denoted by
d
, except when
explicitly mentioned. In this paper, it is always assumed that the discrete metric spaces
under study are
boundedly finite
; i.e., every set included in a ball of finite radius in
D
is finite (note that this is stronger than being locally-finite). This implies that the
metric space is indeed discrete; i.e., every point is isolated. The term
discrete space
will always refer to boundedly finite discrete metric space. A
pointed set
(or a rooted
set) is a pair
(D, o)
, where
D
is a set and
o
a distinguished point of
D
called the
origin
(or the root) of
D
. Similarly, a
doubly-pointed
set is a triple
(D, o1, o2)
, where
o1
and
o2
are two distinguished points of D.
Let
Ξ
be a complete separable metric space called the
mark space
. A
marked
pointed discrete space
is a tuple
(D, o;m)
, where
(D, o)
is a pointed discrete space
and
m
is a function
m:D×D→Ξ
. The mark of a single point
x
may also be defined by
m(x) := m(x, x)
, where the same symbol
m
is used for simplicity. An
isomorphism
(or
rooted isomorphism) between two such spaces
(D, o;m)
and
(D0, o0;m0)
is an isometry
ρ:D→D0
such that
ρ(o) = o0
and
m0(ρ(u), ρ(v)) = m(u, v)
for all
u, v ∈D
. An
isomorphism between doubly-pointed marked discrete spaces is defined similarly. An
isomorphism from a space to itself is called an automorphism.
Most of the examples of discrete spaces in this work are graphs or discrete subsets of
the Euclidean space. More precisely, connected and locally-finite simple graphs equipped
with the graph-distance metric [
2
] are instances of discrete spaces. Similarly, networks;
i.e., graphs equipped with marks on the edges [
2
], are instances of marked discrete
spaces.
Let
D∗
(resp.
D∗∗
) be the set of equivalence classes of pointed (resp. doubly-pointed)
discrete spaces under isomorphism. Let
D0
∗
and
D0
∗∗
be defined similarly for marked
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Unimodular Hausdorff and Minkowski dimensions
discrete spaces with mark space
Ξ
(which is usually given). The equivalence class
containing (D, o),(D, o;m)etc., is denoted by brackets [D, o],[D, o;m], etc.
Every element of
D∗
can be regarded as a boundedly-compact measured metric
space (where the measure is the counting measure). Therefore, the generalization of the
Gromov-Hausdorff-Prokhorov metric in [
37
] defines a metric on
D∗
. By using the results
of [
37
], one can show that
D∗
is a Borel subset of some complete separable metric space.
The proof of this result is skipped for brevity. It can be found in [
7
]. Similarly, one can
show that
D∗∗,D0
∗
and
D0
∗∗
are Borel subsets of some complete separable metric spaces
(see also [
36
]). This enables one to define random pointed discrete spaces, etc., which
are discussed in the next subsection.
2.3 Random Pointed Discrete Spaces
Definition 2.2.
A
random pointed discrete space
is a random element in
D∗
and is
denoted by bold symbols
[D,o]
. Here,
D
and
o
represent the discrete space and the
origin respectively.
In this paper, the probability space is not referred to explicitly
1
. The main reason
is that the notions of dimension, to be defined, depend only on the distribution of the
random object. Also, extra randomness will be considered frequently and it is easier to
forget about the probability space. By an abuse of notation, the symbols
P
and
E
are
used for all random objects, possibly living in different spaces. They refer to probability
and expectation with respect to the random object under consideration.
Note that the whole symbol
[D,o]
represents one random object, which is a random
equivalence class of pointed discrete spaces. Therefore, any formula using
D
and
o
should be well defined for equivalence classes; i.e., it should be invariant under pointed
isomorphisms.
The following convention is helpful throughout.
Convention 2.3.
In this paper, bold symbols are usually used in the random case or
when extra randomness is used. For example,
[D, o]
refers to a deterministic element of
D∗and [D,o]refers to a random pointed discrete space.
Note that the distribution of a random pointed network
[D,o]
is a probability measure
on D∗defined by µ(A) := P[[D,o]∈A]for events A⊆ D∗.
Definition 2.4.
A
random pointed marked discrete space
is a random element in
D0
∗
and is denoted by bold symbols
[D,o;m]
. Here,
D
,
o
and
m
represent the discrete
space, the origin and the mark function respectively.
Most examples in this work are either random rooted graphs (or networks) [
2
] or
point processes (i.e., random discrete subset of
Rk
) and marked point processes that
contain 0, where 0 is considered as the origin. Other examples are also studied by
considering different metrics on such objects.
2.4 Unimodular Discrete Spaces
Once the notion of random pointed discrete space is defined, the definition of uni-
modularity is a straight variant of [
2
]. In what follows, the notation is similarly to [
5
].
Here, the symbol
g[D, o, v]
is used as a short form of
g([D, o, v])
. Similarly, brackets
[·]
are used as a short form of ([·]).
1
Indeed, one may regard
D∗
, equipped with a probability measure, as the canonical probability space. The
last paragraph of Subsection 2.2 ensures that this is a standard probability space, and hence, the classical
tools of probability theory are available. One may also define a random pointed discrete space as a measurable
function from some standard probability space to D∗.
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Unimodular Hausdorff and Minkowski dimensions
Definition 2.5.
A
unimodular discrete space
is a random pointed discrete space,
namely [D,o], such that for all measurable functions g:D∗∗ →R≥0,
E"X
v∈D
g[D,o, v]#=E"X
v∈D
g[D, v, o]#.(2.1)
Similarly, a
unimodular marked discrete space
is a random pointed marked discrete
space [D,o;m]such that for all measurable functions g:D0
∗∗ →R≥0,
E"X
v∈D
g[D,o, v;m]#=E"X
v∈D
g[D, v, o;m]#.(2.2)
Note that the expectations may be finite or infinite.
When there is no ambiguity, the term
g[D, o, v]
is also denoted by
gD(o, v)
or simply
g(o, v)
. The sum in the left (respectively right) side of
(2.1)
is called the
outgoing mass
from o
(respectively
incoming mass into o
) and is denoted by
g+(o)
(respectively
g−(o)
). The same notation can be used for the terms in
(2.2)
. So
(2.1)
and
(2.2)
can be
summarized by
Eg+(o)=Eg−(o).
These equations are called the
mass transport principle
in the literature. The reader
will find further discussion on the mass transport principle and unimodularity in [
2
] and
the examples therein.
As a basic example, every finite metric space
D
, equipped with a random root
o∈D
chosen uniformly, is unimodular. Also, the lattices of the Euclidean space rooted at 0;
e.g.,
[Zk,0]
and
[δZk,0]
, are unimodular. In addition, unimodularity is preserved under
weak convergence, as observed in [13] for unimodular graphs.
The following two examples show that unimodular discrete spaces unify unimodular
graphs and point-stationary point processes. Most of the examples in this work are of
these types.
Example 2.6
(Unimodular Random Graphs)
.
In the case of random rooted graphs and
networks, the concept of unimodularity in Definition 2.5 coincides with that of [
2
] (see
also Remark A.5 of [
7
] regarding the topologies). Therefore, unimodular random graphs
and networks are special cases of unimodular (marked) discrete spaces.
Example 2.7
(Point-Stationary Point Processes)
.Point-stationarity
is defined for point
processes
Φ
in
Rk
such that
0∈Φ
a.s. (see e.g., [
39
]). This definition is equivalent
to
(2.1)
, except that
g
is required to be invariant under translations only (and not under
all isometries). This implies that
[Φ,0]
is unimodular. In addition, by considering the
mark
m(x, y) := y−x
on pairs of points of
Φ
, point-stationarity of
Φ
will be equivalent
to the unimodularity of
[Φ,0; m]
(see also Remark A.5 of [
7
] regarding the topologies).
Note also that Φcan be recovered from [Φ,0; m].
For example, if
Φ
is a stationary point process in
Rk
(i.e., its distribution is invariant
under all translations), with finite intensity (i.e., a finite expected number of points in
the unit cube), then the Palm version of
Φ
is a point-stationary point process, where
the latter is heuristically obtained by conditioning
Φ
to contain the origin (see e.g.,
Section 13 of [
18
] for the precise definition). Also, if
(Xn)n∈Z
is a stochastic process in
Rk
with stationary increments such that
X0= 0
, then the graph of this random walk is
a point-stationary point process in
Rk+1
. The image of this random walk is also point-
stationary provided that it is discrete (i.e., the random walk is transient) and
Xi6=Xj
a.s. for every i6=j. See Subsection 4.3.
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Unimodular Hausdorff and Minkowski dimensions
2.5 Equivariant Process on a Unimodular Discrete Space
In many cases in this paper, an unmarked unimodular discrete space
[D,o]
is given
and various ways of assigning marks to
D
are considered. Intuitively, an equivariant
process on
D
is an assignment of (random) marks to
D
such that the new marked space
is unimodular. Formally, it is
a unimodular marked discrete space
[D0,o0;m]
such that the space
[D0,o0]
,
obtained by forgetting the marks, has the same distribution as [D,o].
In this paper, it is more convenient to work with a disintegrated form of this heuristic,
defined below. It can be proved that the two notions are equivalent, but the proof is
skipped for brevity (this claim is similar to invariant disintegration for group actions). The
easy part of the claim is Lemma 2.12 below. For the other direction, see Proposition B.1
of [7].
In the following, the mark space Ξis fixed as in Subsection 2.2.
Definition 2.8.
Let
D
be a deterministic discrete space which is boundedly-finite. A
marking
of
D
is a function from
D×D
to
Ξ
; i.e., an element of
ΞD×D
. A
random
marking of Dis a random element of ΞD×D.
Definition 2.9.
An
equivariant process Z
with values in
Ξ
is a map that assigns to
every deterministic discrete space
D
a random marking
ZD
of
D
satisfying the following
properties:
(i) Z
is compatible with isometries in the sense that for every isometry
ρ:D1→D2
,
the random marking ZD1◦ρ−1of D2has the same distribution as ZD2.
(ii) For every measurable subset A⊆D0
∗, the following function on D∗is measurable:
[D, o]7→ P[[D, o;ZD]∈A].
In addition, given a unimodular discrete space
[D,o]
, such a map is also called an
equivariant process on D
. In this case, one can also let
Z(·)
be undefined for a class
of discrete spaces, as long as it is defined for almost all realizations of
D
. It is important
that extra randomness be allowed here.
Convention 2.10.
If
D
is clear from the context,
ZD(·)
is also denoted by
Z(·)
for
simplicity.
Note that in the above definition,
D
is deterministic and is not an equivalence class of
discrete spaces. However, for an equivariant process on
[D,o]
, one can define
[D,o;ZD]
as a random pointed marked discrete space with distribution Q(on D0
∗), defined by
Q(A) := Z Z 1A[D, o;m]dPD(m)dµ([D, o]),∀A⊆ D0
∗,(2.3)
where
PD
is the distribution of
ZD
(for every
D
) and
µ
is the distribution of
[D,o]
(note
that only the distribution of
ZD
is important here and it doesn’t matter which probability
space is used for
ZD
). It can be seen that
Q(A)
is indeed well defined and is a probability
measure on
D0
∗
. As mentioned before, the probabilities and expectations to be used
for
ZD
and
[D,o;ZD]
will be denoted by the same symbols
P
and
E
; e.g.,
P[ZD∈B]
,
P[[D, o;ZD]∈A]
and
E[f[D,o;ZD]]
. The symbols
PD
and
Q
will not be used in what
follows.
The following basic examples help to illustrate the definition.
Example 2.11.
By choosing the marks of points (or pairs of points) in an i.i.d. manner,
one obtains an equivariant process. Also, the following periodic marking of
Z
is an
EJP 26 (2021), paper 155. Page 8/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
equivariant process on
Z
: Choose
U∈ {0,1, . . . , n −1}
uniformly at random and let
ZZ(x) := 1
if
x∈nZ+U
and
ZZ(x) := 0
otherwise. Moreover, given a measurable
function
z:D∗∗ →Ξ
, one can define
ZD(u, v) := z[D, u, v ]
, which will be called a
deterministic process here.
Lemma 2.12.
Let
[D,o]
be a unimodular discrete space. If
Z
is an equivariant process
on D, then [D,o;ZD]is also unimodular.
The proof is straightforward and skipped for brevity. The converse of this claim also
holds (see [
7
]). It is important here to assume that the distribution of
ZD
does not
depend on the origin (as in Definition 2.9).
Remark 2.13.
One can easily extend the definition of equivariant processes to allow the
base space to be marked. Therefore, for point-stationary point processes, one can replace
condition
(i)
by invariance under translations only (see Example 2.7). In particular, every
stationary stochastic process on Zkdefines an equivariant process on Zk.
Definition 2.14.
An
equivariant subset S
is the set of points with mark 1 in some
{0,1}
-valued equivariant process. In addition, if
[D,o]
is a unimodular discrete space,
then the intensity of Sin Dis defined by ρD(S) := P[o∈SD].
For example,
SD:= {v∈D: #N1(v)=4}
defines an equivariant subset. Also, let
D=Z
and
SD
be the set of even numbers with probability
p
and the set of odd numbers
with probability
1−p
. Then,
S
is an equivariant subset of
Z
if and only if
p=1
2
(notice
Condition (i) of Definition 2.9).
Lemma 2.15.
Let
[D,o]
be a unimodular discrete space and
S
an equivariant subset.
Then
SD6=∅
with positive probability if and only if it has positive intensity. Equivalently,
SD=Da.s. if and only if ρD(S)=1.
Proof.
The claim is implied by the mass transport principle
(2.2)
for the function
g[D, u, v;S] := 1{v∈S}. The details are left to the reader.
The above lemma is a generalization of similar results in [5] and [2].
2.6 Notes and Bibliographical Comments
The mass transport principle was introduced in [
31
]. The concept of unimodular
graphs was first defined for deterministic transitive graphs in [
12
] and generalized to
random rooted graphs and networks in [2].
Unimodular graphs have many analogies and connections to (Palm versions of) sta-
tionary point processes and point-stationary point processes, as discussed in Example 9.5
of [
2
] and also in [
5
] and [
35
]. As already explained, the framework of unimodular dis-
crete spaces introduced in this section can be regarded as a common generalization of
these concepts.
Special cases of the notion of equivariant processes have been considered in the
literature. The first formulation in Subsection 2.5 is considered in [
2
] for unimodular
graphs. Factors of IID [
40
] are special cases of equivariant processes where the marks of
the points are obtained from i.i.d. marks (Example 2.11) in an equivariant way. Covariant
subsets and covariant partitions of unimodular graphs are defined similarly in [
5
], but no
extra randomness is allowed therein. In the case of stationary (marked) point processes,
the first formulation of Subsection 2.5 is used in the literature. However, the authors
believe that the general formulation of Definition 2.9 is new even in those special cases.
3 The Unimodular Minkowski and Hausdorff Dimensions
This section presents the new notions of dimension for unimodular discrete spaces.
As mentioned in the introduction, the statistical homogeneity of unimodular discrete
EJP 26 (2021), paper 155. Page 9/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
spaces is used to define discrete analogues of the Minkowski and Hausdorff dimensions.
Also, basic properties of these definitions are discussed.
3.1 The Unimodular Minkowski Dimension
Definition 3.1.
Let
[D,o]
be a unimodular discrete space and
r≥0
. An
equivariant r-
covering R
of
D
is an equivariant subset of
D
such that the set of balls
{Nr(v) : v∈RD}
cover
D
almost surely. Here, the same symbol
R
is used for the following equivariant
process (Definition 2.9):
R(v) := RD(v) := r, there is a ball centered at vin the covering,
0,otherwise,
for
v∈D
. Note that an equivariant covering may use extra randomness and is not
necessarily a function of D. This is essential in the following definition.
Let Crbe the set of all equivariant r-coverings. Define
λr:= λr(D) := inf{intensity of Rin D:R∈ Cr},(3.1)
where the intensity is defined in Definition 2.14.
Note that
λr
is non-increasing in terms of
r
. A smaller
λr
heuristically means that a
smaller number of balls per point is needed to cover D. So define
Definition 3.2.
The
upper and lower unimodular Minkowski dimensions
of
D
are
defined by
udimM(D) := decay (λr),
udimM(D) := decay (λr),
as
r→ ∞
. If the decay rate of
λr
exists, define the
unimodular Minkowski dimension
of Dby
udimM(D) := decay (λr).
Here are some first illustrations of the definition.
Example 3.3.
The randomly shifted lattice
Sn:= (2n+ 1)Zk−Un
, where
Un∈
{−n, . . . , n}k
is chosen uniformly, is an equivariant
n
-covering of
Zk
equipped with the
l∞
metric (other metrics can be treated similarly). This implies that
λn≤P[0 ∈Sn] =
(2n+ 1)−k, and hence, udimM(Zk)≥k.
Example 3.4.
If
D
is finite with positive probability, then it can be seen that any
non-empty equivariant subset has intensity at least
E[1/#D]
(use the mass transport
principle when sending mass
1/#D
from every point of the subset to every point of
D
).
This implies that udimM(D) = 0.
Remark 3.5
(Bounding the Minkowski Dimension)
.
In all examples in this work, lower
bounds on the unimodular Minkowski dimension are obtained by constructing explicit
examples of
r
-coverings. Upper bounds can be obtained by constructing disjoint or
bounded coverings, as discussed in Subsection 3.2 below, or by comparison with the
unimodular Hausdorff dimension defined in Subsection 3.3 below (see Theorem 3.22).
3.2 Optimal Coverings for the Minkowski Dimension
Definition 3.6.
Let
[D,o]
be a unimodular discrete space and
r≥0
. If the infimum in
the definition of
λr(3.1)
is attained by an equivariant
r
-covering
S
; i.e.,
P[o∈SD] = λr
,
then Sis called an optimal r-covering for D.
Theorem 3.7.
Every unimodular discrete space has an optimal
r
-covering for every
r≥0.
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Unimodular Hausdorff and Minkowski dimensions
Sketch of the proof.
Let
S1,S2, . . .
be a sequence of equivariant
r
-coverings of
D
such
that
P[o∈Sn]→λr
. By a tightness argument and choosing a subsequence if neces-
sary, one may assume that
[D,o;Sn]
converges weakly, say to
[D,o;S]
, where
S
is an
equivariant subset
S
of
D
(see [
7
] for more details). Since each
Sn
is an
r
-covering,
P[Sn∩Nr(o) = ∅]=0
. It is straightforward to deduce that
P[S∩Nr(o) = ∅]=0
. So by
putting balls of radius
r
on the points of
S
, the root is covered a.s. So Lemma 2.15 implies
that every point is covered a.s.; i.e.,
S
is an
r
-covering. Also, by weak convergence,
P[o∈S] = limnP[o∈Sn] = λr. This implies that Sis an optimal r-covering.
In general, finding an optimal covering is difficult. In some examples, the following is
easier to study.
Definition 3.8.
Let
K < ∞
and
r≥0
. An
r
-covering of
D
is
K-bounded
if each point
of
D
is covered at most
K
times a.s. A sequence
(Rn)n
of equivariant coverings of
D
is
called uniformly bounded if there is K < ∞such that each Rnis K-bounded.
Lemma 3.9. If Ris a K-bounded equivariant r-covering of D, then
1
KP[R(o)6= 0] ≤λr≤P[R(o)6= 0] .(3.2)
So if
(Rn)n
is a sequence of equivariant coverings which is uniformly bounded, with
Rn
an n-covering for each n≥1, then
udimM(D) = decay (P[Rn(o)6= 0]),
udimM(D) = decay (P[Rn(o)6= 0]).
Proof.
The rightmost inequality in (3.2) is immediate from the definition of
λr
. Let
R0
be
another equivariant
r
-covering. Let
g(u, v)=1
if
R0(u) = R(v) = r
and
d(u, v)≤r
. Then
g+(o)≤K1{R0(o)6=0}
and
g−(o)≥1{R(o)6=0}
. Hence by the mass transport principle
(2.2)
,
1
KP[R(o)6= 0] ≤PR0(o)6= 0
and the leftmost inequality in (3.2) then follows from the
definition of λr. The last two equalities follow immediately from (3.2).
Corollary 3.10.
If
R
is an equivariant
disjoint r-covering
of
D
(i.e., the balls used in
the covering are pairwise disjoint a.s.), then it is an optimal r-covering for D.
Example 3.11.
The covering of
Zk
(equipped with the
l∞
metric) constructed in Ex-
ample 3.3 is a disjoint covering. So it is optimal and hence
udimM(Zk) = k
. For
Zk
equipped with the Euclidean metric, one can construct a
3k
-bounded covering similarly
and deduce the same result.
Example 3.12.
Let
Tk
be the
k
-regular tree. For
r≥1
, consider a deterministic covering
of
Tk
by disjoint balls of radius
r
. By choosing
o
in one of these balls uniformly at random,
it can be seen that an equivariant disjoint
r
-covering of
[Tk,o]
is obtained (the proof is
left to the reader). So Corollary 3.10 implies that
λr= 1/#Nr(o)
which has exponential
decay when k≥3. Hence, udimM(Tk) = ∞for k≥3.
Proposition 3.13.
For any point-stationary point process
Φ
on
R
endowed with the
Euclidean metric, by letting p(r) := P[Φ ∩(0, r) = ∅], one has
udimM(Φ) = decay 1
rRr
0p(s)ds≤1∧decay (p(r)) ,
udimM(Φ) = decay 1
rRr
0p(s)ds= 1 ∧decay (p(r)) .
Proof.
Let
r > 0
and
ϕ
be a discrete subset of
R
. Let
Ur
be a random number in
[0, r)
chosen uniformly. For each
n∈Z
, put a ball of radius
r
centered at the largest element
of
ϕ∩[nr +Ur,(n+ 1)r+Ur)
. Denote this random
r
-covering of
ϕ
by
Rϕ
. One can see
that
R
is equivariant under translations (see Remark 2.13). This implies that
R
is an
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Unimodular Hausdorff and Minkowski dimensions
equivariant covering (verifying Condition
(ii)
of Definition I. 2.9 is skipped here). One
has
P[0 ∈RΦ] = P[Φ ∩(0,Ur) = ∅] = 1
rZr
0
P[Φ ∩(0, s) = ∅] ds=: q(r).
Now, since
R
is a 3-bounded covering, Lemma 3.9 implies the two left-hand-side equali-
ties. For all
β < decay (p(r))
, one has
p(r)< r−β
for large enough
r
. So, if in addition,
β < 1
, then
q(r)< cr−β
for some constant
c
, so that
decay (q(r)) ≥β
. Therefore
decay (q(r)) ≥1∧decay (p(r))
. Now, the final equality in the claim is deduced from
q(r)≥p(r)
. Similarly, if
decay (p(r)) <1
, one can deduce
decay (q(r)) ≤decay (p(r))
.
Also,
q(r)≥1
rR1
0p(s)ds
, and hence
decay (q(r)) ≤1
. This implies the first inequality and
completes the proof.
3.3 The Unimodular Hausdorff Dimension
The definition of the unimodular Hausdorff dimension is based on coverings of the
discrete space by balls of possibly different radii. Such a covering can be represented by
an assignment of marks to the points, where the mark of a point
v
represents the radius
of the ball centered at
v
. As mentioned earlier, it is convenient to assume that the radii
are at least 1 (in fact, this condition is technically necessary in what follows). Also, by
convention, if there is no ball centered at
v
, the mark of
v
is defined to be 0. In relation
with this convention, the following notation is used for all discrete spaces
D
and points
v∈D:
Nr(v) := {u∈D:d(v, u)≤r}, r ≥1,
∅, r = 0.
In words, Nr(v)is the closed ball of radius rcentered at v, except when r= 0.
Definition 3.14.
Let
[D,o]
be a unimodular discrete space. An
equivariant (ball-)
covering R
of
D
is an equivariant process on
D
(Definition 2.9) with values in
Ξ :=
{0} ∪ [1,∞)
, such that the family of balls
{NR(v)(v) : v∈D}
covers the points of
D
almost surely. For simplicity,
NR(v)(v)
will also be denoted by
NR(v)
. Also, for
0≤α < ∞
and 1≤M < ∞, let
Hα
M(D) := inf {E[R(o)α] : R(v)∈ {0} ∪ [M, ∞),∀v, a.s.},(3.3)
where the infimum is over all equivariant coverings Rsuch that almost surely, ∀v∈D:
R(v)∈ {0} ∪ [M, ∞)
, and, by convention,
00:= 0
. Note that
Hα
M(D)
is a non-decreasing
function of both αand M.
In the ergodic case,
E[R(o)α]
can be interpreted as the average of
R(·)α
over
the vertices. Also,
P[R(o)>0]
(which is used for defining the unimodular Minkowski
dimension) can be interpreted as the number of balls per point. Ergodicity is however a
special case, and there is no need to assume it in what follows; for more on the matter,
see Example 3.19 and the discussion after it.
Definition 3.15.
Let
[D,o]
be a unimodular discrete space. The number
Hα
1(D)
, defined
in
(3.3)
, is called the
α-dimensional Hausdorff content
of
D
. The
unimodular
Hausdorff dimension of Dis defined by
udimH(D) := sup{α≥0 : Hα
1(D)=0},(3.4)
with the convention that sup ∅= 0.
The key point of assuming equivariance in the above definition is that by Lemma 2.12,
[D,o;R]
is a unimodular marked discrete space. Note also that extra randomness is
allowed in the definition of equivariant coverings. Note also that
0≤ Hα
1(D)≤1,
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Unimodular Hausdorff and Minkowski dimensions
since for the covering by balls of radius 1, one has E[R(o)α]=1.
Examples 3.16 and 3.19 below provide basic illustrations of the unimodular Hausdorff
dimension.
Example 3.16.
If
D
is finite with positive probability, then one can show similarly to
Example 3.4 that
E[R(o)α]≥E[1/#D]
for every
R
, and hence,
udimH(D) = 0
. Also,
for the covering
Sn
of
Zk
constructed in Example 3.3, one has
E[Sn(o)α] = (2n+ 1)α−k
.
If
α < k
, this implies that
Hα
1(Zk) = 0
, and hence,
udimH(Zk)≥k
. The upper bound
udimH(Zk)≤kis implied by Lemma 3.17 below. So udimH(Zk) = k.
Lemma 3.17.
Let
[D,o]
be a unimodular discrete space and
α≥0
. If there exists
c≥0
such that ∀r≥1:#Nr(o)≤crαa.s., then udimH(D)≤α.
Proof.
Let
R
be an arbitrary equivariant covering. For all discrete spaces
D
and
u, v ∈D
, let
gD(u, v)
be 1 if
d(u, v)≤RD(u)
and 0 otherwise. One has
g+(u)=#NR(u)
and
g−(u)≥1
a.s. (since
R
is a covering). By the assumption and the mass transport
principle (2.2), one gets
E[R(o)α]≥1
cE[#NR(o)] = 1
cEg+(o)=1
cEg−(o)≥1
c.
Since Ris arbitrary, one gets Hα
1(D)≥1
c>0, and hence, udimH(D)≤α.
Remark 3.18
(Bounding the Hausdorff Dimension)
.
In most examples in this work, a
lower bound on the unimodular Hausdorff dimension is provided, either by comparison
with the Minkowski dimension (see Subsection 3.4 below), or by explicit construction of a
sequence of equivariant coverings
R1,R2, . . .
such that
E[Rn(o)α]→0
as
n→ ∞
. Note
that this gives
Hα
1(D) = 0
, which implies that
udimH(D)≥α
. Constructing coverings
does not help to find upper bounds for the Hausdorff dimension. The derivation of upper
bounds is mainly discussed in Section 5. The main tools are the mass distribution princi-
ple (Theorem 5.2), which is a stronger form of Lemma 3.17 above, and the unimodular
Billingsley’s lemma (Theorem 5.6).
Example 3.19.
Let
[D,o]
be
[Z,0]
with probability
1
2
and
[Z2,0]
with probability
1
2
. It is
shown below that udimM(D) = udimH(D)=1.
For
n∈N
, the equivariant
n
-covering of Example 3.3 makes sense for
D
and is uni-
formly bounded. One has
P[R(0) >0] = 1
2(n−1+n−2)
. This implies that
udimM(D) =
decay 1
2(n−1+n−2)= 1
. Also, for
α < 1
, one has
E[R(o)α] = 1
2(nα−1+nα−2)→0
as
n→ ∞
. This implies that
Hα
1(D) = 0
for all
α < 1
and hence
udimH(D)≥1
. On the
other hand, for any equivariant covering S, one has
E[S(o)] ≥E[S(o)|D=Z]P[D=Z] = 1
2E[S(o)|D=Z].
Let
c > 2
. The proof of Lemma 3.17 for
[Z,0]
implies that
E[S(o)|D=Z]≥1
c
. This
implies that H1
1(D)≥1
2c>0. So udimH(D)≤1.
Remark 3.20.
The result of this example might seem counterintuitive at first glance as
the union of a filled square and a segment is two dimensional. The number of balls of
radius

required to cover the square dominates the number of balls required to cover
the segment, but in Example 3.19, the situation is reversed: a larger fraction of points is
needed to cover
Z
than
Z2
. This is a consequence of considering large balls and also
counting the number of balls per point. See also Subsection 8.3.
In fact, the following example justifies more clearly why Example 3.19 is one dimensional:
Let
Gn
be the union of a
n×n
square grid (regarded as a graph) and a path of length
n2
sharing a vertex with the grid. To cover
Gn
by balls of radius
r
, a fraction of order
1/r
of
the vertices of
Gn
are needed (as
r
is fixed and
n→ ∞
). So it is not counterintuitive to
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Unimodular Hausdorff and Minkowski dimensions
say that
Gn
is one dimensional asymptotically. Indeed,
Gn
tends to the random graph of
Example 3.19 in the local weak convergence [
2
] as
n→ ∞
(if one chooses the root of
Gn
randomly and uniformly).
Remark 3.21.
In Example 3.19 above, different samples of
D
have different natures
heuristically. This is formalized by saying that
[D,o]
is non-ergodic; i.e., there is an
event
A⊆ D∗
such that the proposition
[D, o]∈A
does not depend on the origin of
D
and
0<P[[D,o]∈A]<1
. In such cases, it is desirable to assign a dimension to every sample
of
D
. In easy examples like Example 3.19, this might be achieved by conditioning. For
instance, in some examples, it is convenient to condition on having infinite cardinality
(which is common, e.g., in branching processes). However, in general, it doesn’t seem
easier to define the dimension of samples separately in a way that is compatible with
the definitions of this paper. In the future work [
6
], the notion of sample dimension is
defined by combining the definitions in this paper with either ergodic decomposition or
conditional expectation. In this work, the reader may focus mainly on the ergodic case,
but it should be noted that the definitions and results do not require ergodicity.
3.4 Comparison of Hausdorff and Minkowski Dimensions
Theorem 3.22 (Minkowski vs. Hausdorff).One has
udimM(D)≤udimM(D)≤udimH(D).
Proof.
The first inequality holds by the definition. For the second one, the definition of
λr(3.1) implies that for every α≥0and r≥1,
inf{E[R(o)α] : Ris an equivariant r-covering}=rαλr.
This readily implies that
Hα
1(D)≤rαλr
for every
r≥1
. So, if
α < decay (λr)
, one gets
Hα
1(D) = 0, and hence, udimH(D)≥α. This implies the claim.
Remark 3.23.
There exist examples in which the inequalities in Theorem 3.22 are strict
(see e.g., Subsections 4.2.2 and 4.4). However, equality holds in most examples. In what
follows, the equality
udimM(D) = udimH(D)
will be referred to as regularity for the
unimodular discrete space D, regarded as a fractal object.
3.5 The Unimodular Hausdorff Size
Consider the setting of Subsection 3.3. For 0≤α < ∞, let
Hα
∞(D) := lim
M→∞ Hα
M(D)∈[0,∞],(3.5)
where Hα
M(D)is defined in (3.3). Note that the limit exists because of monotonicity.
Definition 3.24.
The
unimodular α-dimensional Hausdorff size
of
D
(in short, uni-
modular α-dim H-size of D) is
Mα(D) := (Hα
∞(D))−1.(3.6)
This definition resembles the Hausdorff measure of compact sets. But since
Mα
is
not a measure, the term size is used instead. It can be used to compare unimodular
spaces with equal dimension. The following results gather some elementary properties
of the function Hα
Mand the Hausdorff size.
Lemma 3.25. One has
(i) Hα
1(D)≤ Hα
M(D)≤MαHα
1(D).
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Unimodular Hausdorff and Minkowski dimensions
(ii) Hα
1(D)=0⇔ Hα
∞(D) = 0 ⇔ Mα(D) = ∞.
(iii) If α≥β, then Hα
M(D)≥Mα−βHβ
M(D).
Proof. (i)
. If
R
is an equivariant covering, them
MR
is also an equivariant covering and
satisfies ∀v∈D:MR(v)∈ {0} ∪ [M , ∞)a.s.
(ii). The claim is implied by part (i).
(iii)
. If
R
is an equivariant covering such that
∀v∈D:R(v)∈ {0}∪[M , ∞)
a.s., then
R(o)α≥Mα−βR(o)βa.s.
Lemma 3.26.
If
α < udimH(D)
, then
Hα
∞(D)=0
and
Mα(D) = ∞
. Also, if
α >
udimH(D), then Hα
∞(D) = ∞and Mα(D)=0.
Proof.
For
α < udimH(D)
, one has
Hα
1(D)=0
. So part
(ii)
of Lemma 3.25 implies that
Mα(D) = ∞
. For
α > udimH(D)
, there exists
β
such that
α > β > udimH(D)
. For
this
β
, one has
Hβ
1(D)>0
and part
(iii)
of the same lemma implies that
Hα
M(D)≥
Mα−βHβ
M(D)≥Mα−βHβ
1(D)
. This implies that
Hα
∞(D) = ∞
, which proves the claim.
Remark 3.27.
For
α:= udimH(D)
, the
α
-dim H-size of
D
can be zero, finite or infinite.
The lattice
Zk
provides a case where
Mα(D)
is positive and finite (Proposition 3.29
below). Examples 6.1 and 6.2 provide examples of the infinite and zero cases respectively.
The following propositions provide basic examples of the computation of the Haus-
dorff size.
Proposition 3.28 (0-dim H-size).One has M0(D)=(E[1/#D])−1.
Proof.
As in Example 3.16, one gets
H0
M(D)≥E[1/#D]
. It is enough to show that
equality holds. If
D
is finite a.s., this can be proved by putting a single ball of radius
M∨diam(D)
centered at a point of
D
chosen uniformly at random. Second, assume
D
is infinite a.s. It is enough to construct an equivariant covering
R
such that
P[R(o)>0]
is arbitrarily small. Let
p > 0
be arbitrary and
S
be the Bernoulli equivariant subset
obtained by selecting each point with probability
p
in an i.i.d. manner. For all infinite
discrete spaces
D
and
v∈D
, let
τD(v)
be the closest point of
SD
to
v
(if there is
a tie, choose one of them uniformly at random independently). It can be seen that
τ−1
D(u)
is finite almost surely (use the mass transport principle for
g(v, u) := 1{u=τD(v)}
).
For
u∈SD
, let
R(u) := 1∨diam(τ−1(u))
be the diameter of the Voronoi cell of
u
. For
u∈D\SD
, let
R(u) := 0
. It is clear that
R
is a covering, and in fact, an equivariant
covering. One has
P[R(o)>0] = P[o∈SD] = p
, which is arbitrarily small. So the claim
is proved in this case.
Finally, assume
D
is finite with probability
q
. For all deterministic discrete spaces
D
, let
RD
be one of the above coverings depending on whether
D
is finite or infinite. It
satisfies
P[R(o)>0] = E[1/#D] + p(1 −q)
. Since
p
is arbitrary, the claim is proved.
Proposition 3.29.
For all
δ > 0
, the
k
-dim H-size of the scaled lattice
[δZk,0]
, equipped
with the l∞metric, is equal to (2/δ)k.
Proof.
Let
Sn
be the covering in Example 3.3 scaled by factor
δ
. One has
ESn(o)k=
(nδ)k/(2n+ 1)k
. This easily implies that
Hk
∞(δZk)≤(δ/2)k
. On the other hand, the
proof of Lemma 3.17 shows that
Hk
∞(δZk)≥cδk
, where
c
is any constant such that
rk≥c#Nr(0)
for large enough
r
. It follows that
Hk
∞(δZk)≥(δ/2)k
, and the claim is
proved.
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Unimodular Hausdorff and Minkowski dimensions
3.6 The Effect of a Change of Metric
To avoid confusion when considering two metrics, a pointed discrete space is denoted
by
((D, d), o)
here, where
d
is the metric on
D
and
o
is the origin. Note that if
d0
is
another metric on
D
, then
d0∈RD×D
. So
d0
can be considered as a marking of
D
in the
sense of Definition 2.8 and ((D, d), o;d0)is a pointed marked discrete space.
Definition 3.30.
An
equivariant (boundedly finite) metric
is an
R
-valued equivari-
ant process
d0
such that, for all discrete spaces
(D, d)
,
d0
(D,d)
is almost surely (w.r.t. the
extra randomness) a metric on Dand (D, d0
(D,d))is a boundedly finite metric space.
If in addition,
[(D,d),o]
is a unimodular discrete space, then
[(D,d),o;d0]
is a unimod-
ular marked discrete space by Lemma 2.12. It can be seen that
[(D,d0),o;d]
, obtained
by swapping the metrics, makes sense as a random pointed marked discrete space (see
[
7
] for the measurability requirements). By verifying the mass transport principle
(2.2)
directly, it is easy to show that [(D,d0),o;d]is unimodular.
The following result is valid for both the Hausdorff and the (upper and lower)
Minkowski dimensions.
Theorem 3.31
(Change of Metric)
.
Let
[(D,d),o]
be a unimodular discrete space and
d0
be an equivariant metric. If
d0≤cd+a
a.s., with
c
and
a
constants, then the
dimension of
(D,d0)
is larger than or equal to that of
(D,d)
. Moreover, for every
α≥0
,
Mα(D,d0)≥c−αMα(D,d).
Proof.
The claim is implied by the fact that the ball
Ncr+a((D,d0), v)
contains the ball
Nr((D,d), v)and is left to the reader.
As a corollary, if
1
cd−a≤d0≤cd+a
a.s., then
(D,d0)
has the same unimodular
dimensions as
(D,d)
. Also,
cD
has the same dimension as
D
and
Mα(cD) = c−αMα(D)
.
For instance, this result can be applied to Cayley graphs, which are an important
class of unimodular graphs [
2
]. It follows that the unimodular dimensions of a Cayley
graph do not depend on the generating set. In fact, it will be proved in Subsection 6.6
that these dimensions are equal to the polynomial growth degree of H.
Example 3.32.
Let
[G,o]
be a unimodular graph. Examples of equivariant metrics on
G
are the graph-distance metric corresponding to an equivariant spanning subgraph
(e.g., the drainage network model of Subsection 4.5 below) and metrics generated by
equivariant edge lengths. More precisely, if
l
is an equivariant process which assigns a
positive weight to the edges of every deterministic graph, then one can let
d0(u, v)
be
the minimum weight of the paths that connect
u
to
v
. If
d0
is a metric for almost every
realization of Gand is boundedly-finite a.s., then it is an equivariant metric.
3.7 Dimension of Subspaces
Let
[D,o]
be a unimodular discrete space and
S
be an equivariant subset which is
almost surely nonempty. Lemma 2.15 implies that
P[o∈SD]>0
. So one can consider
[SD,o]
conditioned on
o∈SD
. By directly verifying the mass transport principle
(2.1)
,
it is easy to see that
[SD,o]
conditioned on
o∈SD
is unimodular (see the similar claim
for unimodular graphs in [5]).
Convention 3.33.
For an equivariant subset
S
as above, the unimodular Hausdorff
dimension of
[SD,o]
(conditioned on
o∈SD
) is denoted by
udimH(SD)
. The same
convention is used for the Minkowski dimension, the Hausdorff size, etc.
Theorem 3.34.
Let
[D,o]
be a unimodular discrete space and
S
an equivariant subset
such that SDis nonempty a.s. Then,
EJP 26 (2021), paper 155. Page 16/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
(i) One has
udimH(SD) = udimH(D),
udimM(SD)≥udimM(D),
udimM(SD)≥udimM(D).
(ii)
If
ρ
is the intensity of
S
in
D
, then for every
α≥0
, the
α
-dim H-size of
SD
satisfies
2−αρMα(D)≤ Mα(SD)≤ρMα(D).
Theorem 3.34 is proved below by using the fact that every covering of the larger
set induces a covering of the subset by deleting some balls and then re-centering and
enlarging the remaining balls. This matches the analogous idea in the continuum
setting. The apparently surprising direction of the inequalities is due to the definition of
dimension which implies that having less balls means having larger or equal dimension.
For more on the matter, see the discussion on negative dimension in Subsection 8.3.
Remark 3.35.
Subsection 3.8 below defines a modification
M0
α(D)
of the unimodular
Hausdorff size by considering coverings by arbitrary sets. With this definition, one
has
M0
α(Sd) = ρM0
α(D)
. This can be proved similarly to Theorem 3.34. with the
modification that there is no need to double the radii.
Remark 3.36.
In the setting of Theorem 3.34,
udimM(SD)
can be strictly larger than
udimM(D)
(see, e.g., Subsection 4.4). However, equality holds when
D
is regular (see
Remark 3.23), which immediately follows from Theorems 3.22 and 3.34. Also, equality
is guaranteed if
SD
is a
r
-covering of
D
for some constant
r
. In other words, roughly
speaking, the unimodular dimensions are quasi-isometry invariant (see e.g., [
28
]) and
do not depend on the fine details of the discrete space.
Proof of Theorem 3.34.
The first claim of
(i)
is implied by
(ii)
and Lemma 3.26, and
hence, is skipped. Let
R
be an arbitrary equivariant
r
-covering of
D
. For every
v∈R
,
let
τ(v)
be an element picked uniformly at random in
Nr(v)∩SD
, which is defined only
when
Nr(v)∩SD6=∅
. Let
R0:= {τ(v) : v∈R, Nr(v)∩SD6=∅}
and note that
R0
is a
2r-covering of SD. One has
Po∈R0≤E"X
v
1{v∈R}1{τ(v)=o}#
=E"X
v
1{o∈R}1{τ(o)=v}#≤P[o∈R],
where the equality is by the mass transport principle. This gives
ρλ2r(SD)≤λr(D)
,
which implies the claims regarding the Minkowski dimension.
Now, part
(ii)
is proved. The definition of
Hα
∞(SD)
implies that there exists a sequence
Rn
of equivariant coverings of
SD
such that
Rn(·)∈ {0} ∪ [n, ∞)
for all
n= 1,2, . . .
and
E[Rn(o)α|o∈SD]→ Hα
∞(SD)
. One may extend
Rn
to be defined on
D
by letting
Rn(v) := 0
for
v∈D\SD
. Let
 > 0
be arbitrary and
Bn⊆D
be the union of
N(1+)Rn(v)
for all
v∈D
. Define
R0
n(u) := (1 + )Rn(u)
for
u∈Bn
and
R0
n(u) := 1/
for
u6∈ Bn
. It
is clear that R0
nis an equivariant covering of D. Also,
ER0
n(o)α= (1 + )αE[Rn(o)α] + 1
αP[o6∈ Bn]
=ρ(1 + )αE[Rn(o)α|o∈SD] + 1
αP[o6∈ Bn].(3.7)
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Unimodular Hausdorff and Minkowski dimensions
Since the radii of the balls in
Rn
are at least
n
, one gets that
Bn
includes the
n
-
neighborhood of
SD
. Therefore,
P[o6∈ Bn]≤P[Nn(o)∩SD=∅]
. Since
SD
is non-
empty a.s., this in turn implies that
P[o6∈ Bn]→0
as
n→ ∞
(note that the events
Nn(o)∩SD=∅are nested and converge to the event SD=∅). So (3.7) implies that
lim inf
n→∞
ER0
n(o)α=ρ(1 + )αlim inf
n→∞
E[Rn(o)α|o∈SD] = ρ(1 + )αHα
∞(SD).
Note that the radii of the balls in
R0
n
are at least
n∧(1/)
. Therefore, one obtains
Hα
1/(D)≤ρ(1 + )αHα
∞(SD)
. By letting
→0
, one gets
Hα
∞(D)≤ρHα
∞(SD)
; i.e.,
Mα(SD)≤ρMα(D).
Conversely, let
Rn
be a sequence of equivariant coverings of
D
for
n= 1,2, . . .
such
that
Rn(·)∈ {0} ∪ [n, ∞)
a.s. and
E[Rn(o)α]→ Hα
∞(D)
. Fix
n
in the following. Let
B:= BD:= {v:NRn(v)∩SD6=∅}
. For each
v∈B
, let
τn(v)
be an element chosen
uniformly at random in
NRn(v)∩SD
. For
v6∈ B
, let
τn(v)
be undefined. For
w∈SD
, let
R0
n(w) := 2 max{Rn(v) : v∈τ−1
n(w)}
. It can be seen that
R0
n
is an equivariant covering
of SD. One has
ER0
n(o)α≤2αE"X
v
Rn(v)α1{v∈τ−1
n(o)}#
= 2αE"X
v
Rn(o)α1{o∈τ−1
n(v)}#≤2αE[Rn(o)α],
where the equality is by the mass transport principle. It follows that
ρlim inf
n→∞
ER0
n(o)α|o∈SD≤2αHα
∞(D).
So
ρHα
∞(SD)≤2αHα
∞(D)
. Hence,
Mα(SD)≥2−αρMα(D)
and the claim is proved.
3.8 Covering By Arbitrary Sets
According to Remark 3.35, it is more natural to redefine the Hausdorff size by consid-
ering coverings by finite subsets which are not necessarily balls (as in the continuum
setting). A technical challenge is to define such coverings in an equivariant way. This
will be done at the end of this subsection using the notion of equivariant processes of
Subsection 2.5. Once an equivariant covering
C
is defined (which is an equivariant
collection of finite subsets), one can define the average diameter of sets
U∈C
per point
by
E"X
U∈C
1
#U1{o∈U}diam(U)#.
The same idea is used to redefine Hα
M(D)as follows:
H0
α,M (D) := inf
C(E"X
U∈C
1
#U1{o∈U}M∨1
2diam(U)α#),
where the infimum is over all equivariant coverings
C
. Here, taking the maximum
with
M
is similar to the condition that the subsets have diameter at least
2M
(note
that a ball of radius
M
might have diameter strictly less than
2M
). Finally, define
the
modified unimodular Hausdorff size M0
α(D)
similarly to
(3.6)
. Remark 3.35
shows an advantage of this definition. Also,the reader can verify that
2−αHα
2M(D)≤
H0
α,M (D)≤ Hα
M(D). Therefore,
Mα(D)≤ M0
α(D)≤2αMα(D).
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Unimodular Hausdorff and Minkowski dimensions
This implies that the notion of unimodular Hausdorff dimension is not changed by this
modification. One can also obtain a similar equivalent form of the unimodular Minkowski
dimension. This is done by redefining
λr
by considering equivariant coverings by sets
of diameter at most
2r
. The details are left to the reader. A similar idea will be used in
Subsection 4.1.2 to calculate the Minkowski dimension of one-ended trees.
Finally, here is the promised representation of the above coverings as equivariant
processes (it should be noted that it is not always possible to number the subsets in an
equivariant way and the collection should be necessarily unordered). To show the idea,
consider a covering
C={U1, U2, . . .}
of a deterministic discrete space
D
, where each
Ui
is bounded. For each
Ui
, assign the mark
(Xi,diam(Ui))
to every point of
Ui
, where
Xi∈[0,1]
is chosen i.i.d. and uniformly. Note that multiple marks are assigned to every
point and the covering can be reconstructed from the marks. With this idea, let the mark
space
Ξ
be the set of discrete subsets of
R2
(regard every discrete set as a counting
measure and equip
Ξ
with a metrization of the vague topology). This mark space can be
used to represent equivariant coverings by equivariant processes (for having a complete
mark space, one can extend Ξto the set of discrete multi-sets in R2).
3.9 Notes and Bibliographical Comments
Several definitions and basic results of this section have analogues in the continuum
setting. A list of such analogies is given below. Note however that there is no systematic
way of translating the results in the continuum setting to that of unimodular discrete
spaces. In particular, inequalities are most often, but not always, in the other direction.
The comparison of the unimodular Minkowski and Hausdorff dimensions (Theorem 3.22)
is analogous to the similar comparison in the continuum setting (see e.g., (1.2.3) of [
14
]),
but in the reverse direction. Theorem 3.31, regarding changing the metric, is analogous
to the fact that the ordinary Minkowski and Hausdorff dimensions are not increased
by applying a Lipschitz function. Theorem 3.34 regarding the dimension of subsets
is analogous to the fact that the ordinary dimensions do not increase by passing to
subsets. Note however that equality holds in Theorem 3.34 for the unimodular Hausdorff
dimension (and also for the unimodular Minkowski dimension in most usual examples),
in contrast to the continuum setting.
For point processes (Example 2.7), one can redefine the unimodular Hausdorff
dimension by using dyadic cubes instead of balls. This changes the value of the Hausdorff
size up to a constant factor, and hence, the value of Hausdorff dimension is not changed.
Since dyadic cubes are nested, this simplifies some of the arguments. This approach will
be used in Subsection 7.3.
4 Examples
This section presents a set of examples of unimodular discrete spaces together with
discussions about their dimensions. Recall that the tools for bounding the dimensions
are summarized in Remarks 3.5 and 3.18. As mentioned in Remark 3.18, bounding
the Hausdorff dimension from above usually requires the unimodular mass transport
principle or the unimodular Billingsley lemma, which will be stated in Section 5. So the
upper bounds for some of the following examples are completed later in Subsection 6.1.
4.1 General Unimodular Trees
In this subsection, general results are presented regarding the dimension of unimod-
ular trees with the graph-distance metric. Specific instances are presented later in the
section. It turns out that the number of ends of the tree plays a key role (an
end
in a tree
is an equivalence class of simple paths in the tree, where two such paths are equivalent
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Unimodular Hausdorff and Minkowski dimensions
if their symmetric difference is finite).
It is well known that the number of ends in a unimodular tree belongs to
{0,1,2,∞}
[
2
].
Unimodular trees without end are finite, and hence, are zero dimensional (Example 3.16).
The only thing to mention is that there exists an algorithm to construct an optimal
n
-covering for such trees. This algorithm is similar to the algorithm for one-ended trees,
discussed below, and is skipped for brevity. The case of infinitely many ends will be
studied in Subsection 6.2. The remaining two cases are discussed below.
4.1.1 Unimodular Two-Ended Trees
If
T
is a tree with two ends, then there is a unique bi-infinite path in
T
called its
trunk
.
Moreover, each connected component of the complement of the trunk is finite.
Theorem 4.1.
For all unimodular two-ended trees
[T,o]
endowed with the graph-
distance metric, one has
udimM(T) = udimH(T)=1
. Moreover, if
ρ
is the intensity of
the trunk of T, then the modified 1-dim H-size of Tis M0
1(T) = 2ρ−1.
Proof.
For all two-ended trees
T
, let
ST
be the trunk of
T
. Then,
S
is an equivariant
subset. Therefore, Theorem 3.34 implies that
udimH(T) = udimH(ST)
. Since the trunk
is isometric to
Z
as a metric space, Example 3.16 implies that
udimH(T)=1
. In addition,
Remark 3.35 and Proposition 3.29 imply that M0
1(T) = ρ−1M0
1(Z)=2ρ−1.
The claim concerning the unimodular Minkowski dimension is implied by Corol-
lary 5.10 of the next section, which shows that any unimodular infinite graph satisfies
udimM(G)≥1(this theorem will not be used throughout).
4.1.2 Unimodular One-Ended Trees
Unimodular one-ended trees arise naturally in many examples (see [
2
]). In particular,
the (local weak) limit of many interesting sequences of finite trees/graphs are one-ended
([
3
,
2
]). In terms of unimodular dimensions, it will be shown that unimodular one-ended
trees are the richest class of unimodular trees.
First, the following notation is borrowed from [
5
]. Every one-ended tree
T
can be
regarded as a family tree as follows. For every vertex
v∈T
, there is a unique infinite
simple path starting from
v
. Denote by
F(v)
the next vertex in this path and call it the
parent
of
v
. By deleting
F(v)
, the connected component containing
v
is finite. This set
is denoted by
D(v)
and its elements are called the
descendants
of
v
. The maximum
distance of vto its descendants will be called the height of vand be denoted by h(v).
Theorem 4.2.
If
[T,o]
is a unimodular one-ended tree endowed with the graph-distance
metric, then
udimM(T) = 1 + decay (P[h(o)≥n]) ,(4.1)
udimM(T) = 1 + decay (P[h(o)≥n]) .(4.2)
In addition,
udimH(T)≥decay (P[h(o) = n]) ≥udimM(T).(4.3)
It is not known whether the first inequality in
(4.3)
is always an equality. It should
also be noted that
decay (P[h(o) = n])
can be strictly larger than
1 + decay (P[h(o)≥n])
(see e.g., Subsection 4.2.2), however, they are equal in most usual examples.
The proof of Theorem 4.2 is based on a recursive construction of an optimal covering
by cones, defined below, rather than balls. It is shown below that considering cones
instead of balls does not change the Minkowski dimension. An optimal ball-covering is
also discussed in [7].
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Unimodular Hausdorff and Minkowski dimensions
The
cone
with height
n
at
v∈T
is defined by
Cn(v) := Nn(v)∩D(v)
; i.e., the first
n
generations of the descendants of
v
, including
v
itself. Let
λ0
n
be the infimum intensity of
equivariant coverings by cones of height n. The claim is that
λ0
2n≤λn≤λ0
n.(4.4)
This immediately implies that
udimM(T) = decay (λ0
n),udimM(T) = decay (λ0
n).(4.5)
To prove
(4.4)
, note that any covering by cones of height
n
is also a covering by balls
of radius
n
. This implies that
λn≤λ0
n
. Also, if
S
is a covering by balls of radius
n
, then
{Fn(v) : v∈S}
is a covering by cones of height
2n
. By the mass transport principle
(2.2)
,
one can show that the intensity of the latter is not greater than the intensity of
S
. This
implies that λ0
2n≤λn. So (4.4) is proved.
Lemma 4.3.
For every unimodular one-ended tree
[T,o]
, the output
S
of the following
greedy algorithm is an optimal equivariant covering of Tby cones of height n.
S:= ∅;
while true do
Add all vertices of height nin Tto S;
T:= T\Sv∈SD(v);
end
Note that the algorithm does not finish in finite time, but for each vertex
v
of
T
, it is
determined in finite time whether a cone is put at
v
or not. So the output of the algorithm
is well defined.
Proof.
Let
A
be any equivariant covering of
T
by cones of height
n
. Consider a realization
(T;A)
of
[T;A]
. Let
v
be a vertex such that
h(v) = n
. Since
A
is a covering by cones of
height
n
,
A
should have at least one vertex in
D(v)
(to see this, consider the farthest
leaf from
v
in
D(v)
). Now, for all such vertices
v
, delete the vertices in
A∩D(v)
from
A
and then add
v
to
A
. Let
A1
be the subset of
T
obtained by doing this operation for all
vertices
v
of height
n
. So
A1
is also a covering of
T
by cones of height
n
. Now, remove
all vertices
{v:h(v) = n}
and their descendants from
T
to obtain a new one-ended
tree. Consider the same procedure for the remaining tree and its intersection with
A
.
Inductively, one obtains a sequence of subsets
A=A0, A1, . . .
of
T
such that, for each
i
,
Ai
is a covering of
T
by cones of height
n
which agrees with
ST
on the set of vertices
that are removed from the tree up to step i.
By letting
[T;A]
be random, the above induction gives a sequence of equivariant
subsets
A=A0,A1, . . .
on
T
. It can be seen that the intensity of
A1
is at most that of
A
(this can be verified by the mass transport principle
(2.1)
) and more generally, the
intensity of
Ai+1
is at most that of
Ai
; i.e.,
P[o∈Ai+1]≤P[o∈Ai]
. Also,
limi→∞ Ai=
S
as equivariant subsets of
T
. This implies that
P[o∈A]≥P[o∈S]
, hence,
S
is an
optimal covering by cones of height n.
Lemma 4.4. Under the above setting, one has
P[h(o) mod (n+ 1) = −1] ≤λ0
n≤Phh(o) mod jn
2+1k=−1i.(4.6)
Proof.
The proof of the second inequality in
(4.6)
is based on the construction of the
following equivariant covering. Let
Bn:= {v∈T:h(v) mod n=−1}
and
B0
n:=
{Fn−1(v) : v∈Bn}
. The claim is that
B0
n
is a covering of
T
by cones of height
2n−2
.
Let
v∈T
be an arbitrary vertex. Let
k
be the unique integer such that
(k−1)n−1<
EJP 26 (2021), paper 155. Page 21/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
h(v)≤kn −1
. Let
j
be the first nonnegative integer such that
h(Fj(v)) ≥kn −1
and let
w:= Fj(v)
. One has
0≤j≤n−1
. By considering the longest path in
D(w)
from
w
to the
leaves, one finds
z∈D(w)
such that
h(z) mod n=−1
and
0≤d(w, z)≤n−1
. Therefore
w
(and hence
v
) is a descendant of
Fn−1(z)
. Also,
d(w, F n−1(z)) ≤n−1
. It follows that
d(v, F n−1(z)) ≤2n−2
. So
v
is covered by the cone of height
2n−2
at
Fn−1(z)
. Since
Fn−1(z)∈B0
n
, it is proved that
B0
n
gives a
(2n−2)
-covering by cones. It follows that
λ0
2n−2≤P[o∈B0
n]≤P[o∈Bn]
(where the last inequality can be verified by the mass
transport principle (2.1)). This implies the second inequality in (4.6).
To prove the first inequality in
(4.6)
, let
S
be the optimal covering by cones of height
n
given by the algorithm of Lemma 4.3. Send unit mass from each vertex
v∈S
to the
first vertex in
v, F (v), . . . , F n(v)
which belongs to
Bn+1
(if there is any). So the outgoing
mass from
v
is at most
1{v∈S}
. In the next paragraph, it is proved that the incoming mass
to each
w∈Bn+1
is at least 1. This in turn (by the mass transport principle) implies that
P[o∈S]≥P[o∈Bn+1], which proves the first inequality in (4.6).
The final step consists in proving that the incoming mass to each
w∈Bn+1
is at least
1. If
h(w) = n
, then
w∈S
and the claim is proved. So assume
h(w)> n
. By considering
the longest path from
w
in
D(w)
, one can find a vertex
z
such that
w=Fn+1(z)
and
h(z) = h(w)−(n+ 1)
. This implies that no vertex in
{F(z), . . . , F n(z)}
is in
Bn+1
. So to
prove the claim, it suffices to show that at least one of these vertices or
w
itself lies in
S
. Note that in the algorithm in Lemma 4.3, at each step, the height of
w
decreases
by a value at least 1 and at most
n+ 1
until
w
is removed from the tree. So in the last
step before
w
is removed, the height of
w
is in
{0,1, . . . , n}
. This is possible only if in
the same step of the algorithm, an element of
{F(z), . . . , F n(z), w}
is added to
S
. This
implies the claim and the lemma is proved.
Now, the tools needed to prove the main results are all available.
Proof of Theorem 4.2.
Lemma 4.4 and
(4.5)
imply that the upper and lower Minkowski
dimensions of
T
are exactly the upper and lower decay rates of
P[h(o) mod n=−1]
respectively. So one should prove that these rates are equal to the upper and lower
decay rates of P[h(o)≥n]plus 1.
The first step consists in showing that
P[h(o) = n]
is non-increasing in
n
. To see
this, send unit mass from each vertex
v
to
F(v)
if
h(v) = n
and
h(F(v)) = n+ 1
. Then
the outgoing mass is at most
1{h(v)=n}
and the incoming mass is at least
1{h(v)=n+1}
.
The result then follows by the mass transport principle. This monotonicity implies that
n·P[h(o) mod n=−1] ≥P[h(o)≥n−1]. Similarly, by monotonicity,
n
2P[h(o) mod n=−1] ≤Phh(o) mod n∈ {−1,−2,...,−ln
2m}i
≤Phh(o)≥jn
2ki.
These inequalities conclude the proof of (4.1) and (4.2).
It remains to prove
(4.3)
. The second inequality follows from
(4.1)
and the fact
that
decay (P[h(o) = n]) ≥decay (P[h(o)≥n]) + 1
, which is not hard to see. We now
prove the first inequality. Fix
0<  < α < decay (P[h(o) = n])
. So there is a sequence
n1< n2<···
such that
P[h(o) = ni]< n−α
i
for each
i
. One may assume the sequence is
such that ni≥2ifor each i. Now, for each k∈N, consider the following covering of T:
Rk(v) := 


2(ni−ni−1),if h(v) = niand i > k,
2nk,if h(v) = nk,
0,otherwise.
.
EJP 26 (2021), paper 155. Page 22/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
By arguments similar to Lemma 4.4, it can be seen that
Rk
is indeed a (ball-)covering. It
is claimed that
E[Rk(o)α−]→0
as
k→ ∞
If the claim is proved, then
udimH(T)≥α−
and the proof of (4.3) is concluded. Let c:= 2α−. One has
ERk(o)α−=cnα−
kP[h(o) = nk] + c
∞
X
i=k+1
(ni−ni−1)α−P[h(o) = ni]
≤cn−
k+c
∞
X
i=k+1
(ni−ni−1)α−n−α
i.
Therefore, it is enough to prove that
∞
X
i=1
(ni−ni−1)α−n−α
i<∞.(4.7)
It is easy to see that the maximum of the function
(x−ni−1)α−x−α
over
x≥ni−1
happens at
α
ni−1
and the maximum value is
c0n−
i−1
, where
c0= (α
−1)α−
is a constant.
So the left hand side of
(4.7)
is at most
c0P∞
i=0 n−
i
, which is finite by the assumption
ni≥2i. So (4.7) is proved and the proof is completed.
4.2 Instances of Unimodular Trees
This subsection discusses the dimension of some explicit unimodular trees. More
examples are given in Subsection 4.5, in Section 6, and also in the ongoing work [
6
]
(e.g., uniform spanning forests).
4.2.1 The Canopy Tree
The canopy tree
Ck
with offspring cardinality
k
[
1
] is constructed as follows. Its vertex
set is partitioned in levels
L0, L1, . . .
. Each vertex in level
n
is connected to
k
vertices in
level
n−1
(if
n6= 0
) and one vertex (its parent) in level
n+ 1
. Let
o
be a random vertex
of
Ck
such that
P[o∈Ln]
is proportional to
k−n
. Then,
[Ck,o]
is a unimodular random
tree.
Below, three types of metrics are considered on
Ck
. First, consider the graph-distance
metric. Given
n∈N
, let
S:= {v∈Ck:h(v)≥n}
, where
h(v)
is the height of
v
defined in
Subsection 4.1.2. The set
S
gives an equivariant
n
-covering and
P[o∈S]
is exponentially
small as n→ ∞. So udimM(Ck) = udimH(Ck) = ∞.
Second, for each
n
, let the length of each edge between
Ln
and
Ln+1
be
an
, where
a > 1
is constant. Let
d1
be the resulting metric on
Ck
. Given
r > 0
, let
S1
be the set
of vertices having distance at least
r/a
to
L0
(under
d1
). One can show that
S1
is an
r
-covering of
(Ck, d1)
and
decay (P[o∈S1]) = log k / log a
. Therefore,
udimM(Ck, d1)≥
log k/ log a
. On the other hand, one can see that the ball of radius
an
centered at
o
(under
d1
) has cardinality of order
kn
. One can then use Lemma 3.17 to show that
udimH(Ck, d1)≤log k/ log a. So udimM(Ck, d1) = udimH(Ck, d1) = log k/ log a.
Third, replace
an
by
n!
in the second case and let
d2
be the resulting metric. Then,
the cardinality of the ball of radius
r
centered at
o
has order less than
rα
for every
α > 0
. One can use Lemma 3.17 again to show that
udimH(Ck, d2)≤α
. This implies that
udimM(Ck, d2) = udimH(Ck, d2)=0.
4.2.2 The Generalized Canopy Tree
This example generalizes the canopy tree of Subsection 4.2.1. The goal is to provide an
example where the lower Minkowski dimension, the upper Minkowski dimension and
the Hausdorff dimension are all different when suitable parameters are chosen.
EJP 26 (2021), paper 155. Page 23/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
Fix
p0, p1, . . . > 0
such that
Ppi= 1
. Let
U0,U1, . . .
be an i.i.d. sequence of
random number in
[0,1]
with the uniform distribution. For each
n≥0
, let
Φn:=
1
pn(Z+Un)× {n}
, which is a point process on the horizontal line
y=n
in the plane.
Let
on:= ( 1
pnUn, n)∈Φn
and
Φ := ∪iΦi
. Then,
Φ
is a point process in the plane which is
stationary under horizontal translations. Choose
m
independent of the sequence
(Ui)i
such that P[m=n] = pnfor each n. Then, let o:= om.
Construct a graph
T
on
Φ
as follows: For each
n
, connect each
x∈Φn
to its closest
point (or closest point on its right) in
Φn+1
. Note that
T
is a forest by definition. However,
the next lemma shows that [T,o]is a unimodular tree.
Definition 4.5.
The
generalized canopy tree
with parameters
p0, p1, . . .
is the unimod-
ular tree [T,o]constructed above.
Note that in the case where
pn
is proportional to
k−n
for
k
fixed,
[T,o]
is just
the ordinary canopy tree
Ck
of Subsection 4.2.1. Also, one can generalize the above
construction by letting
Φn
be a sequence of point processes which are (jointly) stationary
under horizontal translations.
Lemma 4.6. One has
(i) [Φ,o], endowed with the Euclidean metric, is a unimodular discrete space.
(ii) Tis a tree a.s. and [T,o]is unimodular.
Proof.
For part
(i)
, it is enough to show that
Φ−o
is a point-stationary point process in
the plane (see Example 2.7). This is skipped for brevity (see [
7
]). The main ingredients
are using stationarity of
Φ
under horizontal translations and the fact that
Φn−on
is
point-stationary (the proof is similar to that of the formula for the Palm version of the
superposition of stationary point processes, e.g., in [47].)
To prove
(ii)
, note that
T
can be realized as an equivariant process on
Φ
(see Defini-
tion 2.9 and Remark 2.13). Therefore, by Lemma 2.12 and Theorem 3.31, it is enough to
prove that
T
is connected a.s. Nevertheless, the same lemma implies that the connected
component
T0
of
T
containing
o
is a unimodular tree. Since it is one-ended, Theorem 3.9
of [
5
] implies that the foils
T0∩Φi
are infinite a.s. By noting that the edges do not
cross (as segments in the plane), one obtains that
T0∩Φi
should be the whole
Φi
; hence,
T0=T. Therefore, Tis connected a.s. and the claim is proved.
Proposition 4.7. The sequence (pn)ncan be chosen such that
udimM(T)<udimM(T)<udimH(T),
where
T
is endowed with the graph-distance metric. Moreover, for any
0≤α≤β≤γ≤
∞, the sequence (pn)ncan be chosen such that
udimM(T)≤α, udimM(T) = β, udimH(T)≥γ.
For example, it is possible to have
udimM(T)=0
and
udimH(T) = ∞
simultaneously.
Proof. T
is a one-ended tree (see Subsection 4.1.2). Assume the sequence
(pn)n
is
non-increasing. So the construction implies that there is no leaf of the tree in
Φn
for
all
n > 0
. Therefore, for all
n≥0
, the height of every vertex in
Φn
is precisely
n
. So by
letting qn:= Pi≥npi, Theorem 4.2 implies that
udimH(T)≥decay (pn),
udimM(T) = 1 + decay (qn),
udimM(T) = 1 + decay (qn).
EJP 26 (2021), paper 155. Page 24/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
For simplicity, assume
0< α
and
γ < ∞
(the other cases can be treated similarly). Define
n0, n1, . . .
recursively as follows. Let
n0:= 0
. Given that
ni
is defined, let
ni+1
be large
enough such that the line connecting points
(ni, n−β
i)
and
(ni+1, n−β
i+1)
intersects the
graph of the function
x−α
and has slope larger than
−n−γ
. Now, let
qni:= n−β
i
for each
i
and define
qn
linearly in the interval
[ni, ni+1]
. Let
pn:= qn−qn+1
. It can be seen that
pn
is non-increasing, decay (qn)≤α,decay (qn) = βand decay (pn)≥γ.
4.2.3 Unimodular Eternal Galton-Watson Trees
Eternal Galton-Watson (
EGW
) trees are defined in [
5
]. Unimodular
EGW
trees (in the
nontrivial case) can be characterized as unimodular one-ended trees in which the
descendants of the root constitute a Galton-Watson tree. Also, unimodularity implies that
the latter Galton-Watson tree is necessarily critical (use the mass transport principle
when sending a unit mass from each vertex to its parent). Here, the trivial case that
each vertex has exactly one offspring is excluded (where the corresponding
EGW
tree is a
bi-infinite path). In particular, the Poisson skeleton tree [
3
] is an eternal Galton-Watson
tree.
Recall that the offspring distribution of a Galton-Watson tree is the probability
measure (p0, p1, . . .)on Z≥0where pnis the probability that the root has noffsprings.
Proposition 4.8.
Let
[T,o]
be a unimodular eternal Galton-Watson tree. If the offspring
distribution has finite variance, then udimM(T) = udimH(T) = 2.
Proof (first part).
Here, it is only proved that
udimM(T) = 2
. The other equality will be
proved in Subsection 6.1. By Kesten’s theorem [
33
] for the Galton-Watson tree formed
by the descendants of the root,
limnnP[h(o)≥n]
exists and is positive. It follows that
decay (P[h(o)≥n]) = 1. So the claim is implied by Theorem 4.2.
4.3 Examples Associated with Random Walks
Let
µ
be a probability measure on
Rk
. Consider the (double-sided) simple random
walk
(Sn)n∈Z
in
Rk
starting from
S0:= 0
such that and the jumps
Sn−Sn−1
are i.i.d. with
distribution
µ
. In this subsection, unimodular discrete spaces are constructed based on
the image and the zero set of this random walk and their dimensions are studied in some
special cases. The graph of the simple random walk will be studied in Subsection 6.4.
4.3.1 The Image of the Simple Random Walk
Assume the random walk is transient; i.e., it visits every given ball only finitely many
times. It follows that the image
Φ = {Sn}n∈Z
is a random discrete subset of
Rk
. If no
point of
Rk
is visited more than once (e.g., when
Sn
is in the positive cone a.s.), then it can
be seen that
Φ
is a point-stationary point process, hence,
[Φ,0]
is a unimodular discrete
space. Hence,
[Φ,0]
is a unimodular discrete space. In the general case, by similar
arguments, one should bias the distribution of
[Φ,0]
by the inverse of the multiplicity of
the origin; i.e., by
1/#{n:Sn= 0}
, to obtain a unimodular discrete space. This claim
can be proved by direct verification of the mass transport principle.
Below, the focus is on the case where the jumps are real-valued and strictly positive.
In this case, Φis actually a point stationary renewal process [24].
Proposition 4.9.
Let
Φ := {Sn}n∈Z
be the image of a simple random walk
S
in
R
starting from S0:= 0. Assume the jumps Sn−Sn−1are positive a.s. Then
(i) udimM(Φ) = decay 1
rE[S1∧r]= 1 ∧decay (P[S1> r]).
(ii) udimM(Φ) = decay 1
rE[S1∧r]≤1∧decay (P[S1> r]).
EJP 26 (2021), paper 155. Page 25/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
(iii) udimH(Φ) ≤1∧decay (P[S1> r]).
(iv) If β:= decay (P[S1> r]) exists, then udimM(Φ) = udimH(Φ) = 1 ∧β.
Proof (first part).
For every
r > 0
, one has
P[Φ ∩(0, r) = ∅] = P[S1≥r]
. So the claims
regarding the Minkowski dimension are direct consequences of Proposition 3.13 and
do not require the i.i.d. assumption. The proofs of the last two claims, will be given in
Subsection 6.1.
The image of the nearest-neighbor simple random walk in
Zk
will be studied in [
6
]. It
will be shown that it has dimension 2 when
k≥2
. Furthermore, a doubling property will
be proved in this case.
As another example, if
[T,o]
is any unimodular tree such that the simple random
walk on
T
is transient a.s., then the image of the (two sided) simple random walk on
T
is
another unimodular tree (after biasing by the inverse of the multiplicity of the root). The
new tree is two-ended a.s., and hence, is 1-dimensional by Theorem 4.1.
4.3.2 Zeros of the Simple Random Walk
Proposition 4.10.
Let
Ψ
be the zero set of the symmetric simple random walk on
Z
with uniform jumps in {±1}. Then, udimM(Ψ) = udimH(Ψ) = 1
2.
Proof.
Represent
Ψ
uniquely as
Ψ := {Sn:n∈Z}
such that
S0:= 0
and
Sn< Sn+1
for
each
n
. Then,
(Sn)n
is another simple random walk and
Ψ
is its image. The distribution of
the jump
S1
is explicitly computed in the classical literature on random walks (using the
reflection principle). In particular, there exist
c1, c2>0
such that
c1r−1
2<P[S1> r]<
c2r−1
2
for every
r≥1
. Therefore, the claim is implied by part
(iv)
of Proposition 4.9
(recall that this part of Proposition 4.9 will be proved later).
4.4 A Subspace with Larger Minkowski Dimension
Let
Φ⊆R
be an arbitrary point-stationary point process. Let
S1
be the first point
of
Φ
on the right of the origin. Assume
β:= decay (P[S1> r])
exists with
β < 1
. Then
Proposition 4.9 gives that udimM(Φ) = β.
Let
α < β < 1
. Consider the intervals defined by consecutive points of
Φ
. In each
such interval, say
(a, b)
, add
d(b−a)αe−1
points so as to split the interval into
d(b−a)αe
equal parts. Let
Φ0
denote the resulting point process (with the points of
Φ
and the
additional points). The assumption
α < β
implies that
E[Sα
1]<∞
. Now, by biasing
the distribution of
Φ0
by
dSα
1e
and changing the origin to a point of
Φ0∩[0, S1)
chosen
uniformly at random, one obtains a point-stationary point process
Ψ
(see Theorem 5 in
[
35
] and also the examples in [
2
]), (it is not a renewal process). The distribution of
Ψ
is
determined by the following equation, where
h
is any measurable nonnegative function:
E[h(Ψ)] = 1
E[dSα
1e]E
X
x∈Φ0∩[0,S1)
h(Φ0−x)
.(4.8)
Proposition 4.11.
Let
Φ
and
Ψ
be as above. Then,
Φ
has the same distribution as an
equivariant subspace of Ψ(conditioned on having the root) and
udimM(Φ) = β > β−α
1−α= udimM(Ψ).
Note that Theorem 3.34 implies that
udimH(Φ) = udimH(Ψ)
. Therefore, the proposi-
tion implies udimM(Ψ) <udimH(Ψ).
EJP 26 (2021), paper 155. Page 26/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
Proof.
Let
A
be the set of newly-added points in
Ψ
, which can be defined by adding
marks from the beginning and is an equivariant subset of
Ψ
. By
(4.8)
, one can verify
that
Ψ\A
conditioned on
06∈ A
has the same distribution as
Φ
(see also Proposition 6
in [35]). Also, by letting c:= E[dSα
1e], (4.8) gives
P[Ψ ∩(0, r) = 0] = 1
cE
X
x∈Φ0∩[0,S1)
1{(Φ0−x)∩(0,r)=∅}

=1
cEdSα
1e1{Φ0∩(0,r)=∅}
=1
cEhdSα
1e1{S1/dSα
1e>r}i.
Now, by the assumption
decay (P[S1> r]) = β
and integration by parts, it is straight-
forward to deduce that
decay (P[Ψ ∩(0, r) = 0]) = (β−α)/(1 −α)
. Therefore, Proposi-
tion 3.13 gives the claim.
Remark 4.12.
The fact that
Ψ
has a smaller Minkowski dimension than
Φ
means that
the tail of the distribution of the jumps (or inter-arrivals) of
Ψ
is heavier than that of the
inter-arrivals of
Φ
. This may look surprising as the inter-arrival times of
Ψ
are obtained
by subdividing those of
Φ
into smaller sub-intervals. The explanation of this apparent
contradiction is of the same nature as that of Feller’s paradox (Section I.4 of [
24
]). It
comes from the renormalization of size-biased sampling: the typical inter-arrival of
Ψ
has more chance to be found in a larger inter-arrival of
Φ
, and this length-biasing
dominates the effect of the subdivision.
4.5 A Drainage Network Model
Practical observations show that large river basins have a fractal structure. For
example, [
30
] discovered a power law relating the area and the height of river basins.
There are various ways to model river basins and their fractal properties in the literature.
In particular, [
45
] formalizes and proves a power law with exponent
3/2
for a specific
model called Howard’s model. Below, the simpler model of [
43
] is studied. One can ask
similar questions for Howard’s model or other drainage network models.
Connect each
(x, y)
in the even lattice
{(x, y)∈Z2:x+ymod 2 = 0}
to either
(x−1, y −1)
or
(x+ 1, y −1)
with equal probability in an i.i.d. manner to obtain a directed
graph
T
. Note that the downward path starting at a given vertex is the rotated graph of
a simple random walk. It is known that
T
is connected and is a one-ended tree (see e.g.,
[45]). Also, by Lemma 2.12, [T,0] is unimodular.
Note that by considering the Euclidean metric on
T
, the Hausdorff dimension of
T
is
2. In the following, the graph-distance metric is considered on T.
Proposition 4.13.
One has
udimM(T) = udimH(T) = 3
2
under the graph-distance
metric.
Proof (first part).
Here, it will be proved that
udimM(T) = 3
2
. The rest of the proof is
postponed to Subsection 6.1. The idea is to use Theorem 4.2. Following [
45
], there are
two backward paths (going upward) in the odd lattice that surround the descendants
D(o)
of the origin. These two paths have exactly the same distribution as (rotated) graphs
of independent simple random walks starting at
(−1,0)
and
(1,0)
, respectively, until they
hit for the first time. In this setting,
h(o)
is exactly the hitting time of these random
walks. So classical results on random walks imply that
P[h(o)≥n]
is bounded between
two constant multiples of
n−1
2
for all
n
. So Theorem 4.2 implies that
udimM(T) = 3
2
.
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Unimodular Hausdorff and Minkowski dimensions
4.6 Self Similar Unimodular Discrete Spaces
This section provides a class of examples of unimodular discrete spaces obtained by
discretizing self-similar sets. Let
l≥1
and
f1, . . . , fl
be similitudes of
Rk
with similarity
ratios
r1, . . . , rl
respectively (i.e.,
∀x, y ∈Rk:|fi(x)−fi(y)|=ri|x−y|
). For every
n≥0
and every string
σ= (j1, . . . , jn)∈ {1, . . . , l}n
, let
fσ:= fj1···fjn
. Also let
|σ|:= n
. Fix a
point
o∈Rk
(one can similarly start with a finite subset of
Rk
instead of a single point).
Let K0:= {o}and Kn+1 := Sjfj(Kn)for each n≥0. Equivalently,
Kn={fσ(o) : |σ|=n}.(4.9)
Recall that if
ri<1
for all
i
, then by contraction arguments,
Kn
converges in the
Hausdorff metric to the attractor of
f1, . . . , fl
(see e.g., Section 2.1 of [
14
]). The attractor
is the unique compact set
K⊆Rk
such that
K=Sifi(K)
. In addition, if the
fi
’s satisfy
the open set condition; i.e., there is a bounded open set
V⊆Rk
such that
fi(V)⊆V
and
fi(V)∩fj(V) = ∅
for each
i, j
, then the Minkowski and Hausdorff dimensions of
K
are
equal to the similarity dimension, which is the unique α≥0such that Prα
i= 1.
The following is the main result of this section. It introduces a discrete analogue of
self-similar sets by scaling the sets Knand taking local weak limits.
Theorem 4.14.
Let
on
be a point of
Kn
chosen uniformly at random, where
Kn
is
defined in (4.9). Assume that
ri=r < 1
for all
i
and the open set condition is satisfied.
Then,
(i) [r−nKn,on]converges weakly to some unimodular discrete space.
(ii)
The unimodular Minkowski and Hausdorff dimension of the limiting space are equal
to α:= log l/|log r|. Moreover, it has positive and finite α-dim H-size.
The proof is given at the end of this subsection. In fact, a point process
Ψ
in
Rk
will
be constructed such that
[r−nKn,on]
converges weakly to
[Ψ, o]
. In addition,
Ψ−o
is
point-stationary. It can also be constructed directly by the algorithm in Remark 4.21
below.
Definition 4.15.
The unimodular discrete space in Theorem 4.14 is called a
self similar
unimodular discrete space.
It should be noted that self similar unimodular discrete spaces depend on the choice
of the initial point oin general.
The following are examples of unimodular self similar discrete spaces. The reader is
also invited to construct a unimodular discrete version of the Sierpinski carpet similarly.
Example 4.16.
If
f1(x) := x/2
and
f2(x) := (1 + x)/2
, then the limiting space is just
Z
.
Similarly, the lattice
Zk
and the triangular lattice in the plane are self similar unimodular
discrete spaces.
Example 4.17
(Unimodular Discrete Cantor Set)
.
Start with two points
K0:= {0,1}
. Let
f1(x) := x/3
and
f2(x) := (2 + x)/3
. Then,
Kn
is the set of the interval ends in the
n
-th
step of the definition of the Cantor set. Here, it is easy to see that the random set
Ψn:= 3n(Kn−on)⊆Z
converges weakly to the random set
Ψ⊆Z
defined as follows:
Ψ := ∪nTn
, where
Tn
is defined by letting
T0:= {0,±1}
and
Tn+1 := Tn∪(Tn±2×3n)
,
where the sign is chosen i.i.d., each sign with probability
1/2
. Note that each
Tn
has the
same distribution as
Ψn
, but the sequence
Tn
is nested. In addition, since
on
is chosen
uniformly,
Ψn
and
Ψ
are point-stationary point processes, and hence
[Ψ,0]
is unimodular
(a deterministic discrete Cantor set exists in the literature which is not unimodular).
Theorem 4.14 implies that udimM(Ψ) = udimH(Ψ) = log 2/log 3.
Example 4.18
(Unimodular Discrete Koch Snowflake)
.
Let
Cn
be the set of points in
the
n
-th step of the construction of the Koch snowflake. Let
xn
be a random point of
EJP 26 (2021), paper 155. Page 28/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
Figure 1: Four ways to attach 3 isometric copies to
Tn
in the construction of the
unimodular discrete Koch snowflake, where each copy is a rotated/translated version of
Tn(relative to Anand Bn). Here, Tnis shown in black.
Cn
chosen uniformly and
Φn:= 3n(Cn−xn)
. It can be seen that
Φn
tends weakly to a
random discrete subset
Φ
of the triangular lattice which is almost surely a bi-infinite
path (note that the cycle disappears in the limit). It can be seen that
Φ
can be obtained
by Theorem 4.14. In this paper,
Φ
is called the
unimodular discrete Koch snowflake
.
Also, Theorem 4.14 implies that udimM(Φ) = udimH(Φ) = log 4/log 3.
In addition,
Φ
can be constructed explicitly as
Φ := ∪nTn
, where
Tn
is a random finite
path in the triangular lattice with distinguished end points
An
and
Bn
defined inductively
as follows: Let
T1:= {A1,B1}
, where
A1
is the origin and
B1
is a neighbor of the origin
in the triangular lattice chosen uniformly at random. For each
n≥1
, given
(Tn,An,Bn)
,
let
(Tn+1,An+1 ,Bn+1)
be obtained by attaching to
Tn
three isometric copies of itself
as shown in Figure 1. There are 4 ways to attach the copies and one of them should be
chosen at random with equal probability (the copies should be attached to
Tn
relative to
the position of Anand Bn). It can be seen that no points overlap.
Remark 4.19.
If the
ri
’s are not all equal, the guess is that there is no scaling of the
sequence
[Kn,on]
that converges to a nontrivial unimodular discrete space (which is
not a single point). This has been verified by the authors in the case
o∈V
. In this
case, by letting
an
be the distance of
on
to its closest point in
Kn
, it is shown that
for any
 > 0
,
P[an/(¯r)n< ]→1
2
and
Pan/(¯r)n>1
→1
2
, where
¯r
is the geometric
mean of
r1, . . . , rl
. This implies the claim (note that the counting measure matters for
convergence; e.g., {0,1
n}does not converge to {0}).
To prove Theorem 4.14, it is useful to consider the following nested version of the sets
Kn
(note that
Kn
is not necessarily contained in
Kn+1
, unless
o
is a fixed point of some
fi
). Let
u1,u2, . . .
be i.i.d. uniform random numbers in
{1, . . . , l}
and
δn:= (un,...,u1)
.
Let
o0
n:= fδn(o)
. Let
ˆ
Kn:= f−1
δnKn=f−1
u1···f−1
unKn
. The chosen order of the indices
in
δn
ensures that
ˆ
Kn⊆ˆ
Kn+1
for all
n
. It is easy to see that
[ˆ
Kn, o]
has the same
EJP 26 (2021), paper 155. Page 29/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
distribution as [r−nKn,o0
n]. For v∈ˆ
Kn, let
wn(v) := #{σ:|σ|=n, fσ(o) = fδn(v)}.
One has
wn(v)≤wn+1(v)
. Note that in the case
o∈V
,
wn(·)=1
and the arguments are
much simpler. The reader can assume this at first reading.
In the following, for
x∈Rk
,
Br(x)
represents the closed ball of radius
r
centered at
xin Rk.
Lemma 4.20. Let ˆ
K:= ∪nˆ
Knand w(v) := limnwn(v)for v∈ˆ
K.
(i) w(·)is uniformly bounded.
(ii) Almost surely, ˆ
Kis a discrete set.
(iii)
The distribution of
[ˆ
K, o]
, biased by
1/w(o)
, is the limiting distribution alluded to in
Theorem 4.14.
Proof.
(i). Assume
fσ1(o)=··· =fσl(o)
and
|σj|=n
for each
j≤l
. Let
D
be a fixed
number such that
V
intersects
BD(o)
. Now, the sets
fσj(V)
for
1≤j≤l
are disjoint and
intersect a common ball of radius
Drn
. Moreover, each of them contains a ball of radius
arn
and each is contained in a ball of radius
brn
(for some fixed
a, b > 0
). Therefore,
Lemma 2.2.5 of [
14
] implies that
l≤(D+2b
a)k=: C
. This implies that
wn(·)≤C
a.s.,
hence w(·)≤Ca.s.
(ii). Let
D
be arbitrary as in the previous part. Assume
f−1
δnfσj(o)∈BD(o)
for
j= 1, . . . , l
. Now, for
j= 1, . . . , l
, the sets
fσj(V)
are disjoint and intersect a common
ball of radius
2Drn
. As in the previous part, one obtains
l≤(2D+2b
a)k
. Therefore,
#ND(o)≤(2D+2b
a)k
a.s. Since this holds for all large enough
D
, one obtains that
ˆ
K
is a
discrete set a.s.
(iii). Note that the distribution of
o0
n
is just the distribution of
on
biased by the
multiplicities of the points in
Kn
. It follows that biasing the distribution of
[ˆ
Kn, o]
by
1/wn(o)
gives just the distribution of
[r−nKn,on]
. The latter is unimodular since
on
is
uniform in
Kn
. So the distribution of
[ˆ
K, o]
biased by
1/w(o)
is also unimodular and
satisfies the claim of Theorem 4.14.
Proof of Theorem 4.14.
Convergence is proved in Lemma 4.20. The rest of the proof is
base on the construction of a sequence of equivariant coverings of
ˆ
K
. In this proof, with
an abuse of notation, the dimension of
ˆ
K
means the dimension of the unimodular space
obtained by biasing the distribution of
ˆ
K
by
1/w(o)
(see Lemma 4.20). Let
D > diam(K)
be given, where
K
is the attractor of
f1, . . . , fl
. Let
m > 0
be large enough so that
diam(Km)< D
. Note that each element in
ˆ
K
can be written as
f−1
δnfσ(o)
for some
n
and
some string
σ
of length
n
. Let
γm
be a string of length
m
chosen uniformly at random
and independently of other variables. For an arbitrary nand a string σof length n, let
Uσ:= f−1
δn+mfσ(Km),
zσ:= f−1
δn+mfσfγm(o).
Note that
Uσ⊆ˆ
K
is always a scaling of
Km
with ratio
r−m
and
zσ∈Uσ
. Now, define
the following covering of ˆ
K:
Rm(v) := Dr−m,if v=zσfor some σ,
0,otherwise.
It can be seen that
Rm
gives an equivariant covering. Also, note that
Rm(o)>0
if and
only if
fσfγm(o) = fδn+m(o)
for some
n
and some string
σ
of length
n
. Let
An,m(o)
be
EJP 26 (2021), paper 155. Page 30/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
the set of possible outcomes for
γm
such that there exists a string
σ
of length
n
such
that the last equation holds. One can see that this set is increasing with
n
and deduce
that
wm(o)≤#An,m(o)≤wn+m(o)
. By letting
w0
m(o) := # ∪nAn,m (o)
, it follows that
wm(o)≤w0
m(o)≤w(o)
. According to the above discussion,
Rm(o)>0
if and only if
γm∈ ∪nAn,m(o). So
P[Rm(o)>0|u0,u1, . . . ] = w0
m(o)rmα.
Therefore, by considering the biasing that makes ˆ
Kunimodular, one gets
E1
w(o)1{Rm(o)>0}=Ew0
m(o)rmα
w(o)≤rmα.(4.10)
Since the balls in the covering have radius Dr−m, one gets udimM(ˆ
K)≥α.
On the other hand, by (4.10) and monotone convergence, one finds that
E1
w(o)1{Rm(o)>0}≥1
2rmα,
for large enough
m
. Similar to the proof of part
(i)
of Lemma 4.20, one can show that the
sequence of coverings
Rm
(for
m= 1,2, . . .
) is uniformly bounded. Therefore, Lemma 3.9
implies that
udimM(ˆ
K) = α
. Moreover, since
E[Rm(o)α/w(o)]
is bounded (by
Dα
), one
can get that Mα(ˆ
K)>0.
Lemma 3.17 will be used to bound the Hausdorff dimension. Let
D > 1
be arbitrary.
Choose
m
such that
r−m≤D < r−m−1
. By Lemma 4.20, there are finitely many points
in
ˆ
K∩BD(o)
. Therefore, one finds
n
such that
ˆ
K∩BD(o) = ˆ
Kn+m∩BD(o)
. It follows
that the sets
{Uσ:|σ|=n}
cover
ˆ
Kn+m
. Now, assume
σ1, . . . , σk
are strings of length
n
such that Uσiare distinct and intersects BD(o). One obtains that
#BD(o)∩ˆ
K≤
k
X
j=1
#BD(o)∩Uσj≤klm=kr−αm ≤kDα.(4.11)
Consider the sets
Vσj:= f−1
δn+mfσj(V)
which are disjoint (since
σj
’s have the same
length). Note that if
 > diam(V∪ {o})
is fixed, then the

-neighborhood of
V
contains
Km
. Therefore, all
Vσj
’s intersect a common ball of radius
D+r−m≤(1 + )D
.
Moreover, each of them contains a ball of radius
ar−m≥arD
and is contained in a ball
of radius
br−m≤bD
(for some
a, b > 0
not depending on
D
). Therefore, Lemma 2.2.5
of [14] implies that k≤((1+)+2b
ar )k. Therefore, (4.11) implies that
#BD(o)∩ˆ
K≤CDα,a.s.
Therefore, Lemma 3.17 implies that
udimH(ˆ
K)≤α
. Moreover, the proof of the lemma
shows that Mα(ˆ
K)<∞. This completes the proof.
Remark 4.21.
Motivated by Examples 4.17 and 4.18, it can be seen that every unimod-
ular self similar discrete space can be constructed by successively attaching copies of a
set to itself. This is expressed in the following algorithm.
4.7 Notes and Bibliographical Comments
Some of the examples in this section, listed below, are motivated by analogous
examples in the continuum setting. In fact, the unimodular dimensions of these examples
are equal to the ordinary dimensions of the analogous continuum examples. This
connection will be discussed further in [6] via scaling limits.
Proposition 4.8 is inspired by the dimension of the Brownian continuum random
tree (see [
29
] or Theorem 5.5 of [
20
]), which is the scaling limit of Galton-Watson trees
EJP 26 (2021), paper 155. Page 31/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
ˆ
K0:= {o};
Let g0be the identity map;
Choose i.i.d. random numbers i1, i2, . . . uniformly in {1, . . . , l};
for n= 1,2, . . . do
let ˆ
Knconsist of lisometric copies of ˆ
Kn−1as follows
ˆ
Kn:=
l
[
j=1
gn−1f−1
infjg−1
n−1(ˆ
Kn−1);
Let gn:= gn−1f−1
in;
end
conditioned to be large. The zero set of the simple random walk (Proposition 4.10) is
analogous to the zero set of Brownian motion. Self-similar unimodular discrete spaces
are inspired by continuum self-similar sets (see e.g., Section 2.1 of [
14
]) as discussed in
Subsection 4.6.
5 The Unimodular Mass Distribution Principle and Billingsley
Lemma
Let
D
be a discrete space and
o∈D
. The
upper and lower (polynomial) volume
growth rates of Dare
growth (#Nr(o)) = lim sup
r→∞
log #Nr(o)/log r,
growth (#Nr(o)) = lim inf
r→∞ log #Nr(o)/log r.
D
has
polynomial growth
if
growth (#Nr(o)) <∞
. These definitions have various other
names in the literature (e.g., mass dimension [
8
], fractal dimension, or growth degree);
volume growth will be used in the present paper since it is common in the context of
graphs and discrete groups.
If the upper and lower volume growth rates are equal, the common value is called the
volume growth rate
of
D
. Note that for
v∈D
, one has
Nr(o)⊆Nr+c(v)
and
Nr(v)⊆
Nr+c(o)
, where
c:= d(o, v)
. This implies that
growth (#Nr(o))
and
growth (#Nr(o))
do
not depend on the choice of the point o.
In various situations in this paper, some weight in
R≥0
will be assigned to each point
of
D
. In these cases, it is natural to redefine the volume growth rate by considering
the weights; i.e., by replacing
#Nr(o)
with the sum of the weights of the points in
Nr(o)
.
This will be formalized below using the notion of equivariant processes. Recall that an
equivariant process should be defined for all discrete spaces
D
. However, if a random
pointed discrete space
[D,o]
is considered, it is enough to define weights in almost every
realization (see Subsection 2.5 for more on the matter). Also, given
D
, the weights are
allowed to be random.
Definition 5.1.
An
equivariant weight function w
is an equivariant process (Defi-
nition 2.5) with values in
R≥0
. For all discrete spaces
D
and
v∈D
, the (random)
value
w(v) := wD(v)
is called the
weight
of
v
. Also, for
S⊆D
, let
w(S) := wD(S) :=
Pv∈Sw(v).
The last equation shows that one could also call wan equivariant measure.
Assume
[D,o]
is a unimodular discrete space (Subsection 2.4). Lemma 2.12 shows
that [D,o;wD]is a random pointed marked discrete space and is unimodular.
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Unimodular Hausdorff and Minkowski dimensions
5.1 Unimodular Mass Distribution Principle
Theorem 5.2 (Mass Distribution Principle).Let [D,o]be a unimodular discrete space.
(i)
Let
α, c, M > 0
and assume there exists an equivariant weight function
w
such that
∀r≥M:w(Nr(o)) ≤crα
, a.s. Then,
Hα
M(D)
defined in (3.3) satisfies
Hα
M(D)≥
1
cE[w(o)].
(ii) If in addition, wD(o)>0with positive probability, then udimH(D)≤α.
Proof.
Let
R
be an arbitrary equivariant covering such that
R(·)∈ {0} ∪ [M, ∞)
a.s. By
the assumption on w,R(o)α≥1
cw(NR(o)) a.s. Therefore,
E[R(o)α]≥1
cE[w(NR(o))] .(5.1)
Consider the independent coupling of
w
and
R
; i.e., for each deterministic discrete
space
G
, choose
wG
and
RG
independently (see Definition 2.9). Then, it can be seen that
the pair
(w,R)
is an equivariant process. So by Lemma 2.12,
[G,o; (w,R)]
is unimodular.
Now, the mass transport principle (2.2) can be used for
[G,o; (w,R)]
. By letting
g(u, v) :=
w(v)1{v∈NR(u)}
, one gets
g+(o) = w(NR(o))
. Also,
g−(o) = w(o)Pu∈D1{o∈NR(u)}≥
w(o)
a.s., where the last inequality follows from the fact that
R
is a covering. Therefore,
the mass transport principle implies that
E[w(NR(o))] ≥E[w(o)]
(recall that by conven-
tion,
NR(o)
is the empty set when
R(o)=0
). So by
(5.1)
, one gets
E[R(o)α]≥1
cE[w(o)]
.
Since this holds for any
R
, one gets that
Hα
M(D)≥1
cE[w(o)]
and the first claim is
proved. If in addition,
w(o)>0
with positive probability, then
E[w(o)] >0
. Therefore,
Hα
1(D)>0and the second claim is proved.
5.2 Unimodular Billingsley Lemma
The main result of this subsection is Theorem 5.6. It is based on Lemmas 5.3 and 5.4
below. Lemma 5.3 is a stronger version of the mass distribution principle (Theorem 5.2).
Lemma 5.3 (An Upper Bound).Let [D,o]be a unimodular discrete space and α≥0.
(i)
If there exist
c≥0
and
w
is an equivariant weight function such that
lim supr→∞ w(Nr(o))/rα≤c, a.s., then Hα
∞(D)≥1
2αcE[w(o)].
(ii) In addition, if wD(o)>0with positive probability, then udimH(D)≤α.
Proof.
Let
c0> c
be arbitrary. The assumption implies that
sup{r≥0 : w(Nr(o)) >
c0rα}<∞
a.s. For
m≥1
, let
Am:= {v∈D:∀r≥m:w(Nr(v)) ≤c0rα}
, which is an
increasing sequence of equivariant subsets. So
lim
m→∞
P[o∈Am] = 1.(5.2)
Let Rbe an equivariant covering such that R(·)∈ {0} ∪ [m, ∞)a.s. One has
E[R(o)α]≥ER(o)α1{NR(o)∩Am6=∅}.(5.3)
If
NR(o)∩Am6=∅
, then
R(o)6= 0
and hence
R(o)≥m
. In the next step, assume that
this is the case. Let
v
be an arbitrary point in
NR(o)∩Am
. By the definition of
Am
,
one gets that for all
r≥m
,
w(Nr(v)) ≤c0rα
. Since
NR(o)(o)⊆N2R(o)(v)
, it follows that
w(NR(o)) ≤w(N2R(o)(v)) ≤2αc0R(o)α. Therefore, (5.3) gives
E[R(o)α]≥1
2αc0Ew(NR(o))1{NR(o)∩Am6=∅}.(5.4)
EJP 26 (2021), paper 155. Page 33/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
Let
g(u, v) := w(v)1{v∈NR(u)}1{NR(u)∩Am6=∅}
. The outgoing mass from the root is
g+(o) = w(NR(o))1{NR(o)∩Am6=∅}
. Also, since there is a ball
NR(u)
that covers
o
a.s., one has
g−(o)≥w(o)1{o∈Am}
a.s. Therefore, the mass transport principle (2.2)
and
(5.4)
imply that
E[R(o)α]≥1
2αc0Ew(o)1{o∈Am}
. This implies that
Hα
m(D)≥
1
2αc0Ew(o)1{o∈Am}
. By using
(5.2)
and letting
m
tend to infinity, one gets
Hα
∞(D)≥
1
2αc0E[w(o)]. Since c0> c is arbitrary, the first claim is proved.
Part
(ii)
is proved by the same argument as the corresponding statement in Theo-
rem 5.2. The proof leverages Lemma 3.26.
Lemma 5.4
(Lower Bounds)
.
Let
[D,o]
be a unimodular discrete space,
α≥0
and
c > 0
.
Let wbe an arbitrary equivariant weight function such that E[w(o)] <∞.
(i) If ∃r0:∀r≥r0:w(Nr(o)) ≥crαa.s., then udimM(D)≥α.
(ii) If growth (w(Nr(o))) ≥αa.s., then udimH(D)≥α.
(iii) If limδ↓0lim inf r→∞ P[w(Nr(o)) ≤δrα]=0, then udimH(D)≥α.
(iv) If decay Ehexp −w(Nn(o))
nαi≥α, then udimM(D)≥α.
Proof.
The proofs of the first two parts are very similar. The second part is proved first.
(ii)
. Let
β
,
γ
and
κ
be such that
γ < β < κ < α
. Fix
n∈N
. Let
S=SD
be the
equivariant subset obtained by selecting each point
v∈D
with probability
1∧(n−βw(v))
(the selection variables are assumed to be conditionally independent given
[D,o;w]
).
Let
Rn(v) = n
if
v∈SD
,
Rn(v)=1
if
Nn(v)∩SD=∅
, and
Rn(v)=0
otherwise.
Then
Rn
is an equivariant covering. It is shown below that
E[Rn(o)γ]→0
. Let
M:= sup{r≥0 : w(Nr(o)) < rκ}. By the assumption, M < ∞a.s. One has
E[Rn(o)γ] = nγP[o∈SD] + P[Nn(o)∩SD=∅]
=nγE1∧n−βw(o)+E
Y
v∈Nn(o)1−(1 ∧n−βw(v))

≤nγ−βE[w(o)] + Eexp −n−βw(Nn(o))
=nγ−βE[w(o)] + Eexp −n−βw(Nn(o))|M < n P[M < n]
+Eexp −n−βw(Nn(o))|M≥nP[M≥n]
≤nγ−βE[w(o)] + exp −nκ−β+P[M≥n],
where the first inequality holds because
1−(1 ∧x)≤e−x
for all
x≥0
. Therefore,
E[Rn(o)γ]→0
when
n→ ∞
. It follows that
udimH(D)≥γ
. Since
γ
is arbitrary, this
implies udimH(D)≥α.
(i)
. Only a small change is needed in the above proof. For
n≥r0
, let
Rn(v) = n
if either
v∈SD
or
Nn(v)∩SD=∅
, and let
Rn(v) = 0
otherwise. Note that
Rn
is a
covering by balls of equal radii. By the same computations and the assumption
M≤r0
,
one gets
P[Rn(o)6= 0] ≤n−βE[w(o)] + exp −nκ−β,
which is of order
n−β
for large
n
. This implies that
udimM(D)≥β
. Since
β
is arbitrary,
one gets udimM(D)≥αand the claim is proved.
(iii)
. Let
β < α
. It will be proved below that under the assumption of
(iii)
, there is a
sequence
r1, r2, . . .
such that
Eexp −r−β
nw(Nrn(o))→0
. If so, by a slight modification
of the proof of part
(ii)
, one can find a sequence of equivariant coverings
Rn
such that
ERn(o)β<∞and (iii) is proved.
EJP 26 (2021), paper 155. Page 34/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
Let
 > 0
be arbitrary. By the assumption, there is
δ > 0
and
r≥1
such that
P[w(Nr(o)) ≤δrα]< . So
Eexp −r−βw(Nr(o)) ≤Eexp −r−βw(Nr(o))|w(Nr(o)) > δrα
+P[w(Nr(o)) ≤δrα]
≤exp(−δrα−β) + .
Note that for fixed

and
δ
as above,
r
can be arbitrarily large. Now, choose
r
large
enough for the right hand side to be at most
2
. This shows that
Eexp −r−βw(Nr(o))
can be arbitrarily small and the claim is proved.
(iv)
. As before, let
Rn(v) = n
if either
v∈SD
or
Nn(v)∩SD=∅
, and let
Rn(v) = 0
otherwise. The calculations in the proof of part (ii) show that
P[Rn(o)6= 0] ≤n−βE[w(o)] + Eexp −n−βw(Nn(o)).
Now, the assumption implies the claim.
Remark 5.5.
The assumption in part
(iii)
of Lemma 5.4 is equivalent to the condition that
there exists a sequence
rn→ ∞
such that the family of random variables
rα
n/w(Nrn(o))
is tight. Also, from the proof of the lemma, one can see that this assumption is equivalent
to
lim inf
n→∞
Eexp −w(Nn(o))
nα= 0.
Theorem 5.6
(Unimodular Billingsley Lemma)
.
Let
[D,o]
be a unimodular discrete metric
space. Then, for all equivariant weight functions
w
such that
0<E[w(o)] <∞
, one has
ess inf growth (w(Nr(o)))≤udimH(D)
≤ess inf growth (w(Nr(o)))
≤growth (E[w(Nr(o))]) .
Proof.
The first inequality is implied by part
(ii)
of Lemma 5.4. For the second inequality,
assume that
growth (w(Nr(o))) < α
with positive probability. On this event, one has
w(Nr(o)) ≤rα
for large
r
; i.e.,
lim suprw(Nr(o))/rα≤1
. Now, Lemma 5.3 implies
that
udimH(D)≤α
. This proves the second inequality. The last claim follows be-
cause
growth (Xn)≤growth (E[Xn])
for any monotone sequence of nonnegative random
variables Xn(see Lemma C.3 of [7]).
Corollary 5.7.
Under the assumptions of Theorem 5.6, if the upper and lower growth
rates of w(Nr(o)) are almost surely constant (e.g., when [D,o;w]is ergodic), then,
growth (w(Nr(o))) ≤udimH(D)≤growth (w(Nr(o))) a.s. (5.5)
In particular, if growth (w(Nr(o))) exists and is constant a.s., then
udimH(D) = growth (w(Nr(o))) .
In fact, without the assumption of this corollary, an inequality similar to
(5.5)
is valid
for the sample Hausdorff dimension of D, which will be studied in [6].
Remark 5.8.
In many examples, in the unimodular Billingsley lemma, it is enough to
take
w
equal to the counting measure, i.e.,
∀v:w(v) = 1
and
w(Nr(o)) = #Nr(o)
.
Analogously, for many natural fractals in the continuum setting, there is a natural mass
measure that can be used in Billingsley’s lemma.
EJP 26 (2021), paper 155. Page 35/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
Remark 5.9.
In fact, the assumption
E[w(o)] <∞
in Theorem 5.6 is only needed for
the lower bound while the assumption
E[w(o)] >0
is only needed for the upper bound.
These assumptions are also necessary as shown below.
For example, assume
Φ
is a point-stationary point process in
R
(see Example 2.7).
For
v∈Φ
, let
w(v)
be the sum of the distances of
v
to its next and previous points
in
Φ
. This equivariant weight function satisfies
w(Nr(v)) ≥2r
for all
r
, and hence
growth (w(Nr(o))) ≥1
. But
udimH(Φ)
can be strictly less than 1 as shown in Subsec-
tion 4.3.1.
Also, the condition that E[wD(o)] >0is trivially necessary for the upper bound.
Corollary 5.10.
Let
[G,o]
be a unimodular random graph equipped with the graph-
distance metric. If
G
is infinite almost surely, then
udimM(G)≥1
and else,
udimM(G) =
udimH(G)=0.
Proof.
If
G
is infinite a.s., then for
wG≡1
, one has
w(Nr(o)) ≥r
for all
r
. So part
(i)
of
Lemma 5.4 implies the first claim. The second claim is implied by Example 3.16 (this
can be deduced from the unimodular Billingsley lemma as well).
Corollary 5.11.
The unimodular Minkowski and Hausdorff dimensions of any unimodu-
lar two-ended tree are equal to one.
This result has already been shown in Theorem 4.1, but can also be deduced from the
unimodular Billingsley lemma directly. For this, let
w(v)
be 1 if
v
belongs to the trunk of
the tree and 0 otherwise.
Problem 5.12.
In the setting of Corollary 5.7, is it always the case that
udimH(D) =
growth (w(Nr(o)))?
The claim of this problem holds in all of the examples in which both quantities are
computed in this work. This problem is a corollary of Problem 5.21 and the unimodular
Frostman lemma (Theorem 7.2) below. Note that there are examples where
growth (·)6=
growth (·)as shown in Subsections 6.3.1 and 6.5.1.
5.3 Bounds for Point Processes
Example 2.7, explains that for point processes containing the origin, unimodularity
is, roughly speaking, equivalent to point-stationarity. To study the dimension of such
processes, the following covering is used in the next results. Let
ϕ
be a discrete subset
of
Rk
equipped with the
l∞
metric and
r≥1
. Let
C:= Cr:= [0, r)k
,
U:= Ur
be a point
chosen uniformly at random in
−C
, and consider the partition
{C+U+z:z∈rZk}
of
Rk
by cubes. Then, for each
z∈rZk
, choose a random element in
(C+U+z)∩ϕ
independently (if the intersection is nonempty). The distribution of this random element
should depend on the set
(C+U+z)∩ϕ
in a translation-invariant way (e.g., choose with
the uniform distribution or choose the least point in the lexicographic order). Let
R=Rϕ
assign the value
r
to the selected points and zero to the other points of
ϕ
. Then,
R
is an
equivariant covering. Also, each point is covered at most
3k
times. So
R
is
3k
-bounded
(Definition 3.8).
Theorem 5.13
(Minkowski Dimension in the Euclidean Case)
.
Let
Φ
be a point-stationary
point process in
Rk
and assume the metric on
Φ
is equivalent to the Euclidean metric.
Then, for all equivariant weight functions wsuch that wΦ(0) >0a.s., one has
udimM(Φ) = decay (E[w(0)/w(Cr+Ur)]) ≤decay (E[w(0)/w(Nr(0))])
≤growth (E[w(Nr(0))]) ,
udimM(Φ) = decay (E[w(0)/w(Cr+Ur)]) ≤decay (E[w(0)/w(Nr(0))])
≤growth (E[w(Nr(0))]) ,
EJP 26 (2021), paper 155. Page 36/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
where Uris a uniformly at random point in −Crindependent of Φand w.
Proof.
By Theorem 3.31, one may assume the metric on
Φ
is the
l∞
metric without
loss of generality. Given any
r > 0
, consider the equivariant covering
R
described
above, but when choosing a random element of
(Cr+Ur+z)∩ϕ
, choose point
v
with probability
wϕ(v)/wϕ(Cr+Ur+z)
(conditioned on
wϕ
). One gets
P[0 ∈R] =
E[w(0)/w(Cr+Ur)]
. As mentioned above,
R
is equivariant and uniformly bounded (for
all
r > 0
). So Lemma 3.9 implies both equalities in the claim. The inequalities are implied
by the facts that w(Cr+Ur)≤w(Nr(0)) and
Ew(0)
w(Nr(0))E[w(Nr(0))] ≥Ehpw(0)i2>0,
which is implied by the Cauchy-Schwartz inequality.
Example 5.14.
The right-most inequalities in the above theorem can be strict. For
example, let
T > 0
be a random number and let
Φ := 1
TZ
. Then
#Nr(0) ∼1 + T/r
. So,
decay (1/#Nr(0)) = 1
, but it might be the case that
E[#Nr(0)] = ∞
. For an ergodic
example, let
1≤Ti∈Z
be i.i.d. with finite mean but infinite variance (for
i∈Z
). In each
interval
[i, i + 1]
, put
Ti−1
equidistant points and let
Φ0
be the union of these points
together with
Z
. Bias the distribution of
Φ0
by
T0
(Definition 2.1) and then translate
Φ0
by moving a random point in
Φ0∩[0,1)
to the origin. Let
Φ
be the resulting point
process. It can be seen that
Φ
is unimodular and point-stationary. Since
ET2
0=∞
, one
gets E[Nr(0)] ≥E[N1(0)] = ∞. But one can show that decay (E[1/Nr(0)]) = 1.
Proposition 5.15.
If
Φ
is a point-stationary point process in
Rk
and the metric on
Φ
is
equivalent to the Euclidean metric, then udimH(Φ) ≤k.
Proof.
One may assume the metric on
Φ
is the
l∞
metric without loss of generality.
Let
C:= [0,1)k
and
U
be a random point in
−C
chosen uniformly. For all discrete
subsets
ϕ⊆Rk
and
v∈ϕ
, let
C(v)
be the cube containing
v
of the form
C+U+z
(for
z∈Zk
) and
wϕ(v) := 1/#(ϕ∩C(v))
. Now,
w
is an equivariant weight function. The
construction readily implies that
w(Nr(o)) ≤(2r+ 1)k
. Moreover, by
w≤1
, one has
E[w(0)] <∞
. Therefore, the unimodular Billingsley lemma (Theorem 5.6) implies that
udimH(Φ) ≤k.
Proposition 5.16.
If
Ψ
is a stationary point process in
Rk
with finite intensity and
Ψ0
is
its Palm version, then
udimM(Ψ0) = udimH(Ψ0) = k
. Moreover, the modified unimodular
Hausdorff size of
Ψ0
, defined in Section 3.8, satisfies
M0
k(Ψ0) = 2kρ(Ψ)
, where
ρ(Ψ)
is
the intensity of Ψ.
Notice that if
Ψ0⊆Zk
, then the claim is directly implied by Theorem 3.34. The
general case is treated below.
Proof.
For the first claim, by Proposition 5.15 and Theorem 3.22, it is enough to prove
that
udimM(Ψ0)≥k
. Let
Ψ0
be a shifted square lattice independent of
Ψ
(i.e.,
Ψ0=Zk+U
,
where
U∈[0,1)k
is chosen uniformly, independently of
Ψ
). Let
Ψ00 := Ψ ∪Ψ0
. Since
Ψ00
is a superposition of two independent stationary point processes, it is a stationary point
process itself. By letting
p:= ρ(Ψ)/(ρ(Ψ) + 1)
, the Palm version
Ψ00
0
of
Ψ00
is obtained
by the superposition of
Ψ0
and an independent stationary lattice with probability
p
(heads), and the superposition of
Zk
and
Ψ
with probability
1−p
(tails). So part
(i)
of Lemma 5.4 implies that
udimM(Ψ00
0)≥k
. Note that
Ψ00
0
has two natural equivariant
subsets which, after conditioning to contain the origin, have the same distributions as
Ψ0
and
Zk
respectively. Therefore, one can use Theorem 3.34 to deduce that
udimM(Ψ0)≥
udimM(Ψ00
0) = k. Therefore, Proposition 5.15 implies that udimH(Ψ0) = udimM(Ψ0) = k.
EJP 26 (2021), paper 155. Page 37/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
Also, by using Theorem 3.34 twice, one gets
M0
k(Ψ0) = pM0
k(Ψ00
0)
and
M0
k(Zk) =
(1−p)M0
k(Ψ00
0)
. Therefore,
M0
k(Ψ0) = p/(1 −p)M0
k(Zk)
. By the definition of
M0
k
, one can
directly show that M0
k(Zk)=2k(see also Proposition 3.29). This implies the claim.
The last claim of Proposition 5.16 suggests the following, which is verified when
k= 1
in the next proposition.
Conjecture 5.17.
If
Φ
is a point-stationary point process in
Rk
which is not the Palm
version of any stationary point process, then Mk(Φ) = 0.
Proposition 5.18. Conjecture 5.17 is true when k= 1.
Proof.
Denote
Φ
as
Φ = {Sn:n∈Z}
such that
S0= 0
and
Sn< Sn+1
for each
n
. Then,
the sequence
Tn:= Sn+1 −Sn
is stationary under shifting the indices (see Example 2.7).
The assumption that
Φ
is not the Palm version of a stationary point process is equivalent
to
E[S1] = ∞
(see [
18
] or Proposition 6 of [
35
]). Indeed, if
E[S1]<∞
, then one could
bias the probability measure by
S1
(Definition 2.1) and then shift the whole process by
−U, where U∈[0, S1]is chosen uniformly and independently.
Since
E[S1] = ∞
, Birkhoff’s pointwise ergodic theorem [
44
] implies that
limn(T1+
··· +Tn)/n =∞
. This in turn implies that
limr#Nr(0)/r = 0
. Therefore, Lemma 5.3
gives that H1
∞(Φ) = ∞; i.e., M1(Φ) = 0.
5.4 Connections to Birkhoff’s Pointwise Ergodic Theorem
The following corollary of the unimodular Billingsley lemma is of independent interest.
Note that the statement does not involve dimension.
Theorem 5.19.
Let
[D,o]
be a unimodular discrete space. For any two equivariant
weight functions
w1
and
w2
such that
P[∃v∈D:w2(v)6= 0] = 1
and
E[w1(o)] <∞
, one
has
growth (w1(Nr(o))) ≤growth (w2(Nr(o))) , a.s.
In particular, if
w1(Nr(o))
and
w2(Nr(o))
have well defined growth rates, then their
growth rates are equal.
Note that the condition E[w1(o)] <∞is necessary as shown in Remark 5.9.
Proof. Let  > 0be arbitrary and
A:= {[D, o]∈ D∗: growth (w1(Nr(o))) >growth (w2(Nr(o))) + }.
It can be seen that
A
is a measurable subset of
D∗
. Assume
P[[D,o]∈A]>0
. Denote by
[D0,o0]
the random pointed discrete space obtained by conditioning
[D,o]
on
A
. Since
A
does not depend on the root (i.e., if
[D, o]∈A
, then
∀v∈D: [D, v]∈A
), by a direct
verification of the mass transport principle (2.1), one can show that
[D0,o0]
is unimodular.
So by using the unimodular Billingsley lemma (Theorem 5.6) twice, one gets
ess inf growth (w1(Nr(o0)))≤udimH(D0)≤ess inf growth (w2(Nr(o0))).
By the definition of
A
, this contradicts the fact that
[D0,o0]∈A
a.s. So
P[[D,o]∈A] = 0
and the claim is proved.
Remark 5.20.
Theorem 5.19 is a generalization of a weaker form of Birkhoff’s pointwise
ergodic theorem as explained below. In the cases where
D
is either
Z
, the Palm version of
a stationary point process in
Rk
or a point-stationary point process in
R
, Birkhoff’s point-
wise ergodic theorem (or its generalizations) implies that
lim w1(Nr(o))/w2(Nr(o)) =
E[w1(0)] /E[w2(0)]
a.s. This is stronger than the claim of Theorem 5.19. Note that
Theorem 5.19 implies nothing about
lim w1(Nr(o))/w2(Nr(o))
. On the other side, note
EJP 26 (2021), paper 155. Page 38/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
that amenability is not assumed in this Theorem, which is a general requirement in the
study of ergodic theorems. However, it will be proved in [
6
] that, roughly speaking, non-
amenability implies
growth (w2(Nr(o))) = ∞
, which makes the claim of Theorem 5.19
trivial in this case. In this case, using exponential gauge functions seems more interest-
ing.
Problem 5.21.
Is it true that for every unimodular discrete space
[D,o]
, the growth
rates
growth (w(Nr(o)))
and
growth (w(Nr(o)))
do not depend on
w
as long as
0<
E[w(o)] <∞?
5.5 Notes and Bibliographical Comments
As already mentioned, the unimodular mass distribution principle and the unimodular
Billingsley lemma have analogues in the continuum setting (see e.g., [
14
]) and are
named accordingly. Note however that there is no direct or systematic reduction to
these continuum results. For instance, in the continuum setting, one should assume that
the space under study is a subset of the Euclidean space, or more generally, satisfies
the bounded subcover property (see e.g., [
14
]). Theorem 5.6 does not require such
assumptions. Note also that the term
growth (w(Nr(o)))
in Theorem 5.6 does not depend
on the origin in contrast to the analogous term in the continuum version. Similar
observations can be made on Theorem 5.2.
6 Examples Continued
This section presents further examples for illustrating the results of the previous
section.
6.1 Remaining Proofs from Section 4
The unimodular Billingsley lemma can be used to complete the computation of the
unimodular Hausdorff dimension in the examples of Section 4. These examples include
Eternal GW trees, the image of a random walk and the drainage network model.
Proof of Proposition 4.8 (second part).
The equality
udimM(T)=2
is proved in Sub-
section 4.2.3. So it remains to prove
udimH(T)≤2
. By the unimodular Billingsley
lemma, it is enough to show that
E[#Nn(o)] ≤cn2
for a constant
c
. Recall from Sub-
section 4.1.2 that
F(v)
represents the parent of vertex
v
and
D(v)
denotes the subtree
of descendants of
v
. Write
Nn(o) = Y0∪Y1∪ ··· ∪ Yn
, where
Yn:= Nn(o)∩D(o)
and
Yi:= Nn(o)∩D(Fn−i(o)) \D(Fn−i−1(o))
for
0≤i < n
. By the explicit construction of
EGW
trees in [
5
],
Yn
is a critical Galton-Watson tree up to generation
n
. Also, for
0≤i<n
,
Yi
has the same structure up to generation
i
, except that the distribution of the first
generation is size-biased minus one (i.e.,
(npn+1)n
with the notation of Subsection 6.3.2).
So the assumption of finite variance implies that the first generation in each
Yi
has finite
mean, namely
m0
. Now, one can inductively show that
E[#Yn] = n
and
E[#Yi] = im0
, for
0≤i<n. It follows that E[#Nn(o)] ≤(1 + m0)n2and the claim is proved.
Proof of Proposition 4.9 (second part).
In Subsection 4.3.1, it is proved that
udimM(Φ)
≥1∧decay (P[S1> r])
. So part
(iv)
is implied by part
(iii)
, which is proved below. Since
Φ
is a point-stationary point process in
R
(see Subsection 4.3.1), Proposition 5.15 implies
that
udimH(Φ) ≤1
. Now, assume
decay (P[S1> r]) < β
. Then, there exists
c > 0
such
that
P[S1> r]> cr−β
for all
r≥1
. This implies that there exists
C < ∞
and a random
number
r0>0
such that for all
r≥r0
, one has
#Nr(o)≤Crβlog log r
a.s. (for the
proof, see [
7
] or Theorem 4 of [
26
]). Therefore, the unimodular Billingsley lemma
EJP 26 (2021), paper 155. Page 39/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
(Theorem 5.6) implies that
udimH(Φ) ≤β+
for every
 > 0
, which in turn implies that
udimH(Φ) ≤β.
Example 6.1
(Infinite H-Size)
.
In Proposition 4.9, assume that
P[S1> r] = 1/log r
for
large enough
r
. Then, part
(iii)
of the proposition implies that
udimH(Φ) = 0
. However,
since Φis infinite a.s., it has infinite 0-dim H-size (Proposition 3.28).
Example 6.2
(Zero H-Size)
.
In Proposition 4.9, assume
P[S1> r]=1/r
for large enough
r
. Then, part
(iii)
of the proposition implies that
udimH(Φ) = 1
. Since
E[S1] = ∞
,
Φ
is
not the Palm version of any stationary point process (see Proposition 5.18). Therefore,
Proposition 5.18 implies that M1(Φ) = 0.
Proof of Proposition 4.13 (second part).
The equality
udimM(T) = 3
2
is proved in Sub-
section 4.5. So it remains to prove
udimH(T)≤3
2
. To use Billingsley’s lemma, an upper
bound on
E[#Nn(o)]
is derived. Let
ek,l := # F−k(Fl(o)) \F−(k−1)(Fl−1(o))
be the
number of descendants of order
k
of
Fl(o)
which are not a descendant of
Fl−1(o)
(for
l= 0
, let it be just
#F−k(o)
). One has
#Nn(o) = Pk,l ek,l 1{k+l≤n}
. It can be seen that
E[ek,l]
is equal to the probability that two independent paths of length
k
and
l
starting
both at
o
do not collide at another point. Therefore,
E[ek,l]≤c(k∧l)−1
2
for some
c
and
all
k, l
. This implies that (in the following,
c
is updated at each step to a new constant
without changing the notation)
E
X
k,l≥0
ek,l1{k+l≤n}
≤bn
2c
X
k=0
ck−1
2(n−k)≤cn bn
2c
X
k=0
k−1
2≤cn3
2.
The above inequalities imply that
E[#Nn(o)] ≤cn3
2
for some
c
and all
n
. Therefore, the
unimodular Billingsley lemma (Theorem 5.6) implies that
udimH(T)≤3
2
. So the claim is
proved.
6.2 General Unimodular Trees Continued
The following is a direct corollary of Theorem 4.2 and the unimodular Billingsley
lemma. Since the statement does not involve dimension, it is of independent interest
and believed to be new.
Corollary 6.3.
For every unimodular one-ended tree
[T,o]
and every equivariant weight
function w, almost surely,
decay (P[h(o) = n]) ≤growth (w(Nr(o))) ≤growth (E[w(Nr(o))]) .
The rest of this subsection is focused on unimodular trees with infinitely many ends.
Proposition 6.4.
Let
[T,o]
be a unimodular tree with infinitely many ends such that
E[deg(o)] <∞. Then Thas exponential volume growth a.s. and udimH(T) = ∞.
In fact, the graph-distance metric on
T
can be replaced by an arbitrary equivariant
metric. This will be proved in [6].
The following proof uses the definitions and results of [
2
], but they are not recalled
for brevity.
Proof of Proposition 6.4.
By Corollary 8.10 of [
2
],
[T,o]
is non-amenable (this will be
discussed further in [
6
]). So Theorem 8.9 of [
2
] implies that the critical probability
pc
of
percolation on
T
is less than one with positive probability. In fact, it can be shown that
pc<1
a.s. (if not, condition on the event
pc= 1
to get a contradiction). For any tree,
pc
is equal to the inverse of the branching number. So the branching number is more than
one, which implies that the tree has exponential volume growth. Finally, the unimodular
Billingsley lemma (Theorem 5.6) implies that udimH(T) = ∞.
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Unimodular Hausdorff and Minkowski dimensions
The following example shows that the Minkowski dimension can be finite.
Example 6.5.
Let
T
be the 3-regular tree. Split each edge
e
by adding a random number
le
of new vertices and let
T0
be the resulting tree. Let
ve
be the middle vertex in this
edge (assuming
le
is always odd) and assign marks by
m0(ve) := le
. Assume that the
random variables
le
are i.i.d. If
E[le]<∞
, then one can bias the probability measure and
choose a new root to obtain a unimodular marked tree, namely
[T,o;m]
(see Example 9.8
of [2] or [35]). It will be shown below that udimM(T)may be finite.
Let
R
be an arbitrary equivariant
r
-covering of
T
. Consider the set of middle vertices
Ar:= {v∈T:m(v)≥r}
. Since these vertices have pairwise distance at least
r
, they
belong to different balls in the covering. So, by the mass transport principle, one can
show that
ρ(R)≥ρ(Ar)
, where
ρ(·) = P[o∈ ·]
denotes the intensity. On the other
hand, let
S
be the equivariant subset of vertices with degree 3. Send unit mass from
every point of
Ar
to its two closest points in
S
. Then the mass transport principle
implies that
2ρ(Ar)=3ρ(S)P[le≥r]
. Hence,
ρ(R)≥3
2ρ(S)P[le≥r]
. This gives that
udimM(T)≤decay (P[le≥r])
, which can be finite. In fact, if
decay (P[le≥r])
exists,
Proposition 6.6 below implies that udimM(T) = decay (P[le≥r]).
The following proposition gives a lower bound on the Minkowski dimension.
Proposition 6.6.
Let
[T,o]
be a unimodular tree with infinitely many ends and without
leaves. Let
S
be the equivariant subset of vertices of degree at least 3. For every
v∈S
,
let
w(v)
be the sum of the distances of
v
to its neighbors in
S
. If
E[w(o)α]<∞
, then
udimM(T)≥α.
The proof is based on the following simpler result. This will be used in Subsec-
tion 6.3.3 as well.
Proposition 6.7.
Let
[T,o]
be a unimodular tree such that the degree of every vertex is
at least 3. Let
d0
be an equivariant metric on
T
. Let
w(v) := Pud0(v, u)
, where the sum
is over the 3 neighbors of
v
which are closest to
v
under the metric
d0
. If
E[w(o)α]<∞
,
then udimM(T,d0)≥α.
Proof.
Define
w0(v) := Pud0(u, v)α
, where the sum is over the three closest neighbors
of
v
. It is enough to assume that
d0
is generated by equivariant edge lengths since
increasing the edge lengths does not increase the dimension (by Theorem 3.31). By the
same argument, it is enough to assume
d0(u, v)≥1
for all
u∼v
. Then, it can be seen
that there exists a constant c, that depends only on α, such that w0(Nr(v)) ≥crαfor all
v∈T
and
r≥0
(see Lemma C.5 in [
7
]). Also, the assumption implies that
E[w0(o)] <∞
.
So Lemma 5.4 implies that udimM(T3,d0)≥αand the claim is proved.
Proof of Proposition 6.6.
For
v∈S
, let
w0(v) := Pud(u, v)α
, where the sum is over
the neighbors of
v
in
S
. For
v∈T\S
, if
u1
and
u2
are the two closest points of
S
to
v
, let
g(v, ui) := d(ui, v)α−1
and
w0(v) := g(v, u1) + g(v, u2)
. The assumption implies
that
E[w0(o)] <∞
(use the mass transport principle for
g
defined above). Similarly to
Proposition 6.7, there exists
c=c(α)
, such that
w0(Nr(v)) ≥crα
for all
v∈T
and
r≥0
(see Lemma C.5 in [7]) and the claim is proved.
6.3 Instances of Unimodular Trees Continued
6.3.1 A Unimodular Tree With No Volume Growth Rate
Recall the generalized canopy tree
[T,o]
from Subsection 4.2.2. Here, it is shown that
growth (T)6= growth (T)
if the parameters are suitably chosen. Similarly, it provides an
example where the exponential growth rate does not exist. The existence of unimodular
trees without exponential growth rate is already proved in [
49
], but with a more difficult
construction.
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Unimodular Hausdorff and Minkowski dimensions
Choose the sequence
(pn)n
in the definition of
[T,o]
such that
pn=c2−qn
and
Pnpn= 1
, where
c
is constant and
q0≤q1≤ ···
is a sequence of integers. In this
case,
T
is obtained by splitting the edges of the canopy tree by adding new vertices or
concatenating them, depending only on the level of the edges. It can be seen that if
v
is a
vertex in the
n
-th level of
T
, then the number of descendants of
v
is
(p0+···+pn)/pn
. It
follows that
growth (T) = decay (pn)
and
growth (T) = decay (pn)
. So, by choosing
(pn)n
appropriately,
T
can have no polynomial (or exponential) volume growth rate. This
proves the claim. Note also that the unimodular Billingsley lemma and Theorem 4.2
imply that udimH(T) = growth (T)here.
6.3.2 Unimodular Galton-Watson Trees
Here, it is shown that the unimodular Galton-Watson tree [
2
] is infinite dimensional (note
that this tree differs from the Eternal Galton-Watson tree of Subsection 4.2.3 which is
a directed tree). Consider an ordinary Galton-Watson tree with offspring distribution
µ= (p0, p1, . . .)
, where
µ
is a probability measure on
Z≥0
. The unimodular Galton-Watson
tree
[T,o]
has a similar construction with the difference that the offspring distribution
of the origin is different from that of the other vertices: It has for distribution the
size-biased version ˆµ= ( n
mpn)n, where mis the mean of µ(assumed to be finite).
In what follows, the trivial case
p1= 1
is excluded. If
m≤1
, then
T
is finite a.s.;
i.e., there is extinction a.s. Therefore,
udimH(T)=0
. So assume the supercritical case,
namely
m > 1
. If
p0>0
, then
T
is finite with positive probability. So
udimH(T)=0
for
the same reason. Nevertheless, one can condition on non-extinction as follows.
Proposition 6.8.
Let
[T,o]
be a supercritical unimodular Galton-Watson tree condi-
tioned on non-extinction. Then, udimM(T) = udimH(T) = ∞.
Proof.
The result for the Hausdorff dimension follows from the unimodular Billings-
ley lemma (Theorem 5.6) and the Kesten-Stigum theorem [
34
], which implies that
limn#Nn(o)m−n
exists and is positive a.s. Computing the Minkowski dimension is more
difficult. By part
(iv)
of Lemma 5.4, it is enough to prove that
E(1 −n−α)#Nn(o)
has
infinite decay rate for every
α≥0
. Denote by
[e
T,e
o]
the Galton-Watson tree with the
same parameters. Using the fact that
#Nn(o)
is stochastically larger than
#Nn−1(e
o)
,
one gets that it is enough to prove the last claim for [e
T,e
o].
For simplicity, the proof is given for the case
p0= 0
only. By this assumption, the
probability of extinction is zero. The general case can be proved with similar arguments
and by using the decomposition theorem of supercritical Galton-Watson trees (see e.g.,
Theorem 5.28 of [
41
]). In fact, the following proof implies the general claim by the
fact that the trunk, conditioned on non-extinction, is another supercritical unimodular
Galton-Watson tree. The latter can be proved similarly to the decomposition theorem.
Let
f(s) := Pnpnsn
be the generating function of
µ
. By classical results of the theory
of branching processes, for all
s≤1
,
Esdn(e
o)=f(n)(s)
, where
dn(˜
o) := #Nn(e
o)−
#Nn−1(e
o)
and
f(n)
is the
n
-fold composition of
f
with itself. Let
a > 0
be fixed and
g(s) := as
−s+a+1
(such functions are frequently used in the literature on branching
processes; see, e.g., [
4
]). One has
f(0) = g(0) = 0
,
f(1) = g(1) = 1
,
f0(1) = m > 1
,
g0(1) = (1 + a)/a
, and
f
is convex. Therefore,
a
can be chosen large enough such that
f(s)≤g(s)for all s∈[0,1]. So
f(n)(s)≤g(n)(s) = ans
an+ (a+ 1)n(1 −s),
where the last equality can be checked by induction. Therefore,
f(n)(1 −n−α)≤an
an+n−α(a+ 1)n.
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Unimodular Hausdorff and Minkowski dimensions
It follows that
decay f(n)(1 −n−α)=∞
. Therefore, the above discussion gives that
E(1 −n−α)#Nn(o)has infinite decay rate and the claim is proved.
6.3.3 The Poisson Weighted Infinite Tree
The Poisson Weighted Infinite Tree (
PWIT
) is defined as follows (see e.g., [
3
]). It is a
rooted tree
[T,o]
such that the degree of every vertex is infinite. Regarding
T
as a
family tree with progenitor
o
, the edge lengths are as follows. For every
u∈T
, the set
{d(u, v) : vis an offspring of u}
is a Poisson point process on
R≥0
with intensity function
xk
, where
k > 0
is a given integer. Moreover, for different vertices
u
, the corresponding
Poisson point processes are jointly independent. It is known that the
PWIT
is unimodular
(notice that although each vertex has infinite degree, the
PWIT
is boundedly-finite as a
metric space). See for example [3] for more details.
Proposition 6.9. The PWIT satisfies udimM(PWIT) = udimH(PWIT) = ∞.
Proof.
Denote the neighbors of
o
by
v1, v2, . . .
such that
d(o, vi)
is increasing in
i
. It is
straightforward that all moments of
d(o, v3)
are finite. Therefore, Proposition 6.7 implies
that udimM(T) = ∞(see also Lemma C.5 in [7]). This proves the claim.
6.4 The Graph of the Simple Random Walk
As in Subsection 4.3, consider the simple random walk
(Sn)n∈Z
in
Rk
, where
S0= 0
and the increments
Sn−Sn−1
are i.i.d. The graph of the random walk
(Sn)n∈Z
is
Ψ := {(n, Sn) : n∈Z} ⊆ Rk+1
. It can be seen that
Ψ
is a point-stationary point process,
and hence, [Ψ,0] is unimodular (see Subsection 4.3.1).
Since
#Ψ ∩[−n, n]k+1 ≤2n+ 1
, the mass distribution principle (Theorem 5.2) implies
that
udimH(Ψ) ≤1
. In addition, if
S1
has finite first moment, then the strong law of large
numbers implies that
limn1
nSn=E[S1]
. This implies that
lim infn1
n#Ψ∩[−n, n]k+1>
0
. Therefore, the unimodular Billingsley lemma (Theorem 5.6) implies that
udimH(Ψ) ≥1
.
Hence,
udimH(Ψ) = 1
. This matches the result of [
38
] that the macroscopic dimension
of the graph of the Brownian motion is 1, while its microscopic dimension is
3/2
when
k= 1 (see Subsection 8.1).
Below, the focus is on the case k= 1 and on the following metric:
d((x, y),(x0, y0)) := max{p|x−x0|,|y−y0|}.(6.1)
Theorem 3.31 implies that, by considering this metric, unimodularity is preserved and
dimension is not decreased. Under this metric, the ball
Nn(0)
is
Ψ∩[−n2, n2]×[−n, n]
. It
is straightforward that
Z2
has volume growth rate 3 and also Minkowski and Hausdorff
dimension 3 under this metric.
Proposition 6.10.
If the jumps are
±1
uniformly, under the metric
(6.1)
, the graph
Ψ
of
the simple random walk satisfies udimM(Ψ) = udimH(Ψ) = 2.
Proof.
Let
n∈N
. The ball
Nn(0)
has at most
2n2+ 1
elements. So the mass distribution
principle (Theorem 5.2) implies that
udimH(Ψ) ≤2
. For the other side, let
C
be the
equivariant disjoint covering of
Z2
by translations of the rectangle
[−n2, n2]×[−n, n]
(similar to Example 3.11). For each rectangle
σ∈ C
, select the right-most point in
σ∩Ψ
and let
S=SΨ
be the set of selected points. By construction,
S
gives an
n
-covering of
Ψ
and it can be seen that it is an equivariant covering. Let
σ0
be the rectangle containing
the origin. By construction,
0∈S
if and only if it is either on a right-edge of
σ0
or on
a horizontal edge of
σ0
and the random walk stays outside
σ0
. The first case happens
with probability
1/(2n2+ 1)
. By classical results concerning the hitting time of random
walks, one can obtain that the probability of the second case lies between two constant
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Unimodular Hausdorff and Minkowski dimensions
multiples of
n−2
. It follows that
P[0 ∈S]
lies between two constant multiples of
n−2
.
Therefore, udimM(Ψ) ≥2. This proves the claim.
6.5 Other Self Similar Unimodular Spaces
In this subsection, two examples are presented which have some kind of self-similarity
heuristically, but do not fit into the framework of Subsection 4.6.
6.5.1 Unimodular Discrete Spaces Defined by Digit Restriction
Let
J⊆Z≥0
. For
n≥0
, consider the set of natural numbers with expansion
(anan−1. . .
a0)
in base 2 such that
ai= 0
for every
i6∈ J
. Similarly to the examples in Subsection 4.6,
one can shift this set randomly and take a limit to obtain a unimodular discrete space.
This can be constructed in the following way as well: Let
T0:= {0}
. If
n∈J
, let
Tn+1 := Tn∪(Tn±2×2n)
, where the sign is chosen i.i.d., each sign with probability
1/2. If n6∈ J, let Tn+1 := Tn. Finally, let Ψ := ∪nTn.
The upper and lower asymptotic densities of
J
in
Z≥0
are defined by
d(J) :=
lim supn1
nJnand d(J) := lim infn1
nJn, where Jn:= #J∩ {0, . . . , n}.
Proposition 6.11. Almost surely,
udimH(Ψ) = udimM(Ψ) = growth (#Nn(o)) = d(J),
udimM(Ψ) = growth (#Nn(o)) = d(J).
In particular, this provides another example of a unimodular discrete space where
the (polynomial) volume growth rate does not exist.
Proof.
Let
n≥0
be given. Cover
Tn
by a ball of radius
2n
centered at the minimal
element of
Tn
. By the same recursive definition, one can cover
Tn+1
by either 1 or 2
balls of the same radius. Continuing the recursion, an equivariant
2n
-covering
Rn
is
obtained. It is straightforward to see that
P[Rn(o)>0] = 2−Jn
. Since these coverings
are uniformly bounded (Definition 3.8), Lemma 3.9 implies that
udimM(Ψ) = d(J)
and
udimM(Ψ) = d(J). One has
#Tm= 2Jm.(6.2)
This implies that
#N2n(o)≤2Jn+1
. One can deduce that
growth (#Nn(o)) ≤d(J)
. So
the unimodular Billingsley lemma (Theorem 5.6) gives
udimH(Ψ) ≤d(J)
. This proves
the claim.
6.5.2 Randomized Discrete Cantor set
This subsection proposes a unimodular discrete analogue of the random Cantor set,
recalled below. Let
0≤p≤1
and
b > 1
. The random Cantor set in
Rk
[
32
] (see also [
14
])
is defined by
Λk(b, p) := ∩nEn
, where
En
is defined by the following random algorithm:
Let
E0:= [0,1]k
. For each
n≥0
and each
b
-adic cube of edge length
b−n
in
En
, divide it
into
bk
smaller
b
-adic cubes of edge length
b−n−1
. Keep each smaller
b
-adic cube with
probability
p
and delete it otherwise independently from the other cubes. Let
En+1
be
the union of the kept cubes. It is shown in Section 3.7 of [
14
] that
Λk(b, p)
is empty for
p≤b−kand otherwise, has dimension k+ logbpconditioned on being non-empty.
For each
n≥0
, let
Kn
be the set of lower left corners of the
b
-adic cubes forming
En
.
It is easy to show that Kntends to Λk(b, p)a.s. under the Hausdorff metric.
Proposition 6.12.
Let
K0
n
denote the random set obtained by biasing the distribution
of Knby #Kn(Definition 2.1). Let o0
nbe a point chosen uniformly at random in K0
n.
(i) [bnK0
n,o0
n]converges weakly to some unimodular discrete space [ˆ
K,ˆ
o].
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Unimodular Hausdorff and Minkowski dimensions
(ii) If p<b−k, then ˆ
Kis finite a.s., hence, udimH(ˆ
K)=0a.s.
(iii) If p≥b−k, then ˆ
Kis infinite a.s. and
udimH(ˆ
K) = udimM(ˆ
K) = k+ logbp, a.s.
Note that in contrast to the continuum analogue [
32
], for
p=b−k
, the set is non-empty
and even infinite, though still zero dimensional. Also, for
p<b−k
the set is non-empty as
well.
To prove the above proposition, the following construction of
ˆ
K
will be used. First,
consider the usual nested sequence of partitions
Πn
of
Zk
by translations of the cube
{0, . . . , bn−1}k
, where
n≥0
. To make it stationary, shift each
Πn
randomly as follows.
Let
a0, a1, . . . ∈ {0,1, . . . , b −1}k
be i.i.d. uniform numbers and let
Un=Pn
i=0 aibi∈Zk
.
Shift the partition
Πn
by the vector
Un
to form a partition denoted by
Π0
n
. It is easy to
see that Π0
nis a nested sequence of partitions.
Lemma 6.13.
Let
(Π0
n)n
be the stationary nested sequence of partitions of
Zk
defined
above. For each
n≥0
and each cube
C∈Π0
n
that does not contain the origin, with
probability
1−p
(independently for different choices of
C
), mark all points in
C∩Zk
for
deletion. Then, the set of the unmarked points of
Zk
, pointed at the origin, has the same
distribution as [ˆ
K,ˆ
o]defined in Proposition 6.12.
Proof of Lemma 6.13.
Let
Φ
be the set of unmarked points in the algorithm. For
n≥0
,
let
Cn
be the cube in
Π0
n
that contains the origin. It is proved below that
Cn∩Φ
has the
same distribution as bn(K0
n−on). This implies the claim.
Let
An⊆[0,1]k
be the set of possible outcomes of
o0
n
. One has
#An=bkn
. For
v∈An
,
it is easy to see that the distribution of
bn(K0
n−on)
, conditioned on
o0
n=v
, coincides
with the distribution of
Cn∩Φ
conditioned on
Cn=bn([0,1)k−v)
. So it remains to
prove that P[o0
n=v] = PCn=bn([0,1)k−v), which is left to the reader.
Here is another description of
ˆ
K
. The nested structure of
SnΠ0
n
defines a tree as
follows. The set of vertices is
SnΠ0
n
. For each
n≥0
, connect (the vertex corresponding
to) every cube in
Π0
n
to the unique cube in
Π0
n+1
that contains it. This tree is the canopy
tree (Subsection 4.2.1) with offspring cardinality
N:= bk
, except that the root (the
cube
{0}
) is always a leaf. Now, keep each vertex with probability
p
and remove it with
probability
1−p
in an i.i.d. manner. Let
T
be the connected component of the remaining
graph that contains the root. Conditioned on the event that
T
is infinite,
ˆ
K
corresponds
to the set of leaves in the connected component of the root.
Proof of Proposition 6.12.
The unimodular Billingsley lemma is used to get an upper
bound on the Hausdorff dimension. For this
E[#Nbn(o)]
is studied. Consider the tree
[T,o]
defined above and obtained by the percolation process on the canopy tree with
offspring cardinality
N:= bk
. Let
C
be any cube in
Π0
i
that does not contain the origin.
Note that the subtree of descendants of
C
in the percolation cluster (conditioned on
keeping
C
) is a Galton-Watson tree with binomial offspring distribution with parameters
(N, p)
. Classical results on branching processes say
Eh#C∩ˆ
K|Π0
ii=pmi
, where
m:= pbk. So the construction implies that
Eh#Cn∩ˆ
Ki= 1 + p(N−1) mn−1+mn−2+· · · + 1.
For
m > 1
, the latter is bounded by
lmn
for some constant
l
not depending on
n
. Note
that
Nbn(o)
is contained in the union of
Cn
and
3k−1
other cubes in
Π0
n
. It follows
that
E[#Nbn(o)] ≤l0mn
, where
l0=l+ (3k−1)p
. So the unimodular Billingsley lemma
EJP 26 (2021), paper 155. Page 45/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
(Theorem 5.6) implies that
udimH(ˆ
K)≤k+ logbp
. The claim for
m= 1
and
m < 1
are
similar.
Consider now the Minkowski dimension. As above, we assume
m > 1
and the
proofs for the other cases are similar. Let
n≥0
be given. By considering the partition
Π0
n
by cubes, one can construct a
bn
-covering
Rn
as in Theorem 5.13. This covering
satisfies
P[Rn(o)≥0] = Eh1/#(Cn∩ˆ
K)i
. Let
[T0,o0]
be the eternal Galton-Watson
tree of Subsection 4.2.3 with binomial offspring distribution with parameters
(N, p)
. By
regarding
T0
as a family tree, it is straightforward that
[T,o]
has the same distribution
as the part of
[T0,o0]
, up to the generation of the root (see [
5
] for more details on
eternal family trees). Therefore, Lemma 5.7 of [
5
] implies that
Eh1/#(Cn∩ˆ
K)i=
m−nP[h(o0)≥n]
. Since
m > 1
,
P[h(o0)≥n]
tends to the non-extinction probability of
the descendants of the root, which is positive. By noticing the fact that the radii of the
balls are
bn
and the covering is uniformly bounded, one gets that
udimM(ˆ
K) = logbm=
k+ logbp.
Finally, it remains to prove that
ˆ
K
is infinite a.s. when
p=b−k
. In this case, consider
the eternal Galton-Watson tree
[T0,o0]
as above. Proposition 6.8 of [
5
] implies that the
generation of the root is infinite a.s. This proves the claim.
6.6 Cayley Graphs
As mentioned in Subsection 3.6, the dimension of a Cayley graph depends only on
the group and not on the generating set. The following result connects it to the volume
growth rate of the group. Note that Gromov’s theorem [
27
] implies that the polynomial
growth degree exists and is either an integer or infinity.
Theorem 6.14.
For every finitely generated group
H
with polynomial growth degree
α∈[0,∞], one has udimM(H) = udimH(H) = α. Also, if α < ∞, then Mα(H)<∞.
Proof.
First, assume
α < ∞
. The result of Bass [
10
] implies that there are constants
c, C > 0
such that
∀r≥1 : crα<#Nr(o)≤Crα
, where
o
is an arbitrary element of
H
.
So the mass distribution principle (Theorem 5.2) and part
(i)
of Lemma 5.4 imply that
udimM(H) = udimH(H) = α
. In addition,
(5.1)
in the proof of Theorem 5.2 implies that
Hα
M(H)≥1/C for all M≥1, which implies that Mα(H)≤C < ∞.
Second, assume
α=∞
. The result of [
50
] shows that for any
β < ∞
,
#Nr(o)> rβ
for
sufficiently large
r
. Therefore, part
(i)
of Lemma 5.4 implies that
udimM(H)≥β
. Hence,
udimM(H) = udimH(H) = ∞and the claim is proved.
It is natural to expect that
Mα(H)>0
as well, but only a weaker inequality will be
proved in Proposition 7.18.
6.7 Notes and Bibliographical Comments
The proof of Proposition 6.4 was suggested by R. Lyons. Bibliographical comments
on some of the examples discussed in this section can be found at the end of Section 4.
The example defined by digit restriction (Subsection 6.5.1) is inspired by an example in
the continuum setting (see e.g., Examples 1.3.2 of [
14
]). The randomized discrete Cantor
set (Subsection 6.5.2) is inspired by the random cantor set (see e.g., Section 3.7 of [
14
]).
7 Frostman’s Theory
This section provides a unimodular version of Frostman’s lemma and some of its
applications. In a sense to be made precise later, this lemma gives converses to the mass
distribution principle. It is a powerful tool in the theoretical analysis of the unimodular
Hausdorff dimension. For example, it is used in this section to derive inequalities for the
EJP 26 (2021), paper 155. Page 46/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
dimension of product spaces and embedded spaces (Subsections 7.4.2 and 7.4.3). It is
also the basis of many of the results in [6].
7.1 Unimodular Frostman Lemma
The statement of the unimodular Frostman lemma requires the definition of weighted
Hausdorff content. The latter is based on the notion of equivariant weighted collections
of balls as follows. For this, the following mark space is needed. Let
Ξ
be the set of
functions
c:R≥0→R≥0
which are positive in only finitely many points; i.e.,
c−1((0,∞))
is a finite set. Remark 7.9 below defines a metric on
Ξ
, so that the notion of
Ξ
-valued
equivariant processes (Definition 2.9) is well defined. Such a process
c
is called an
equivariant weighted collection of balls 2
. Consider a unimodular discrete space
[D,o]
with distribution
µ
. For
v∈D
, the reader can think of the value
cr(v) := c(v)(r)
,
if positive, to indicate that there is a ball in the collection, with radius
r
, centered at
v
,
and with
cost
(or weight)
cr(v)
. Note that extra randomness is allowed in the definition.
A ball-covering
R
can be regarded a special case of this construction by letting
cr(v)
be
1 when r=R(v)and 0 otherwise.
Definition 7.1.
Let
f:D∗→R
be a measurable function and
M≥1
. An equivariant
weighted collection of balls cis called a (f, M )-covering if
∀v∈D:f(v)≤X
u∈DX
r≥M
cr(u)1{v∈Nr(u)}, a.s., (7.1)
where f(v) := f[D, v]for v∈D. For α≥0, define
ξα
M(f) := inf (E"X
r
cr(o)rα#:cis a (f, M )-covering),
ξα
∞(f) := lim
M→∞ ξα
M(f).
It is straightforward that every equivariant ball-covering of Definition 3.14 gives a
(1,1)
-covering, where the first
1
is regarded as the constant function
f≡1
on
D∗
. This
gives (see also Conjecture 7.4 below) that for all M≥1,
ξα
M(1) ≤ Hα
M(D).(7.2)
Also, by considering the case
cM(v) := f(v)∨0
, one can see that if
f∈L1(D∗, µ)
,
then
ξα
M(f)≤MαE[f(o)∨0] <∞.
In the next theorem, to be consistent with the setting of the paper, the following notation
is used:
w(u) := w([D, u])
for
u∈D
, and
w(Nr(v)) = Pu∈Nr(v)w(u)
. Also, recall
that a deterministic equivariant weight function is given by a measurable function
w:D∗→R≥0(see Example 2.11).
Theorem 7.2
(Unimodular Frostman Lemma)
.
Let
[D,o]
be a unimodular discrete space,
α≥0and M≥1.
(i)
There exists a bounded measurable weight function
w:D∗→R≥0
such that
E[w(o)] = ξα
M(1) and almost surely,
∀v∈D,∀r≥M, w(Nr(v)) ≤rα.(7.3)
2
The term ‘weighted’ refers to the weighted sums in Definition 7.1 and should not be confused with
equivariant weight functions of Definition 5.1
EJP 26 (2021), paper 155. Page 47/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
(ii)
In addition, if either
udimH(D)< α
or
α= udimH(D)
and
Mα(D)<∞
, then
w[D,o]6= 0 with positive probability.
The proof is given later in this subsection.
Remark 7.3.
One can show that if
w
is an equivariant weight function satisfying
(7.3)
,
then
E[w(o)] ≤ξα
M(1)
and
E[w(o)h(o)] ≤ξα
M(h)
. Therefore, the (deterministic) weight
function
w
given in the unimodular Frostman lemma is a maximal equivariant weight
function satisfying
(7.3)
(it should be noted that such maximal functions are not unique
in general). The proof is similar to that of the mass distribution principle (Theorem 5.2)
and is left to the reader.
Conjecture 7.4. One has Hα
M(D) = ξα
M(1).
Here are a few comments on this conjecture. An analogous equality holds in the
continuum setting (see 2.10.24 in [
23
]). Lemma 7.8 below proves a weaker inequality. It
can be seen that the conjecture holds for
Zk
(with the
l∞
metric) and for the non-ergodic
example of Example 3.19. In the former case, this is obtained by considering the constant
weight function
w(·)≡M
2M+1 k
, which satisfies the claim of the unimodular Frostman
lemma. The latter case is similar by letting w[Z,0] := M
2M+1 and w[Z2,0] := 0.
Remark 7.5.
The unimodular Frostman lemma implies that, in theory, the mass distribu-
tion principle (Theorem 5.2) is enough for bounding the Hausdorff dimension from above.
However, there are very few examples in which the function
w
given by the unimodular
Frostman lemma can be explicitly computed (in some of the examples, a function
w
satisfying only
(7.3)
can be found; e.g., for two-ended trees). Therefore, in practice, the
unimodular Billingsley lemma is more useful than the mass distribution principle.
The following lemma is needed to prove Theorem 7.2.
Lemma 7.6.
The function
ξα
M:L1(D∗, µ)→R
is continuous. In fact, it is
Mα
-Lipschitz;
i.e., |ξα
M(f1)−ξα
M(f2)| ≤ MαE[|f1(o)−f2(o)|].
Proof.
Let
c
be an equivariant weighted collection of balls satisfying
(7.1)
for
f1
. Intu-
itively, add a ball of radius
M
at each point
v
with cost
|f2(v)−f1(v)|
. More precisely,
let
c0
r(v) := cr(v)
for
r6=M
and
c0
M(v) := cM(v) + |f2(v)−f1(v)|
. This definition implies
that c0satisfies (7.1) for f2. Also,
ξα
M(f2)≤E"X
i
c0
i(o)iα#=E"X
r
cr(o)iα#+MαE[|f2(o)−f1(o)|].
Since
c
is arbitrary, one obtains
ξα
M(f2)≤ξα
M(f1) + MαE[|f2(o)−f1(o)|]
, which implies
the claim.
Proof of Theorem 7.2. The theorem is a special case of Proposition 7.7 which is proved
below.
Proposition 7.7.
In the setting of Theorem 7.2, let
h∈L1(D∗, µ)
be any given function
such that
h > 0
a.s. Then, one can replace the condition
E[w(o)] = ξα
M(1)
in Theorem 7.2
by E[w(o)h(o)] = ξα
M(h), and the conclusions of the theorem are valid.
Proof.
It is easy to see that
ξα
M(tf) = tξα
M(f)
for all
f
and
t≥0
and also
ξα
M(f1+f2)≤
ξα
M(f1) + ξα
M(f2)
for all
f1, f2
. Let
h∈L1(D∗, µ)
be given. By the Hahn-Banach theorem
(see Theorem 3.2 of [
46
]), there is a linear functional
l:L1(D∗, µ)→R
such that
l(h) = ξα
M(h)
and
−ξα
M(−f)≤l(f)≤ξα
M(f)
, for all
f∈L1
. Since
l
is sandwiched between
two functions which are continuous at
0
and are equal at
0
(since
ξα
M(0) = 0
), one gets
that
l
is continuous at 0. Since
l
is linear, this implies that
l
is continuous. Since the dual
EJP 26 (2021), paper 155. Page 48/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
of
L1(D∗, µ)
is
L∞(D∗, µ)
, one obtains that there is a function
w∈L∞(D∗, µ)
such that
l(f) = E[f(o)w(o)]
, for all
f∈L1
. Note that if
f≥0
, then
ξα
M(−f) = 0
and so
l(f)≥0
.
This implies that
w(o)≥0
a.s. (otherwise, let
f(o) := 1{w(o)<0}
to get a contradiction).
Consider a version of
w
which is nonnegative everywhere. The claim is that
w
satisfies
the requirements.
Let
r≥M
be fixed. For all discrete spaces
D
, let
S:= SD:= {v∈D:w(Nr(v)) > rα}
.
By the definition of SD, one has
Ew(Nr(o))1{o∈S}≥rαP[o∈S].(7.4)
Moreover, if
P[o∈S]>0
, then the inequality is strict. Let
fr(v) := #Nr(v)∩S
. By the
mass transport principle for the function (v, u)7→ w(u)1{v∈S}1{u∈Nr(v)}, one gets
Ew(Nr(o))1{o∈S}=E[w(o)#Nr(o)∩S]
=E[w(o)fr(o)]
=l(fr)
≤ξα
M(fr)
≤rαP[o∈S],
where the last inequality is implied by considering the following weighted collection of
balls for
fr
: put balls of radius
r
with cost 1 centered at the points in
S
. More precisely,
let
cr(v) := 1{v∈S}
and
cs(v) := 0
for
s6=r
. It is easy to see that this satisfies
(7.1)
for
fr
, which implies the last inequality by the definition of
ξα
M(·)
. Thus, equality holds
in
(7.4)
. Hence,
P[o∈S]=0
; i.e.,
w(Nr(o)) ≤rα
a.s. Lemma 2.15 implies that almost
surely,
∀v∈D:w(Nr(v)) ≤rα
. So the same holds for all rational
r≥M
simultaneously.
By monotonicity of
w(Nr(v))
w.r.t.
r
, one gets that the latter almost surely holds for
all
r≥M
as desired. Also, one has
E[w(o)h(o)] = l(h) = ξα
M(h)
. Thus,
w
satisfies the
desired requirements.
To prove
(ii)
, assume
Mα(D)<∞
. By Lemma 3.25, one has
Hα
M(D)>0
. So
Lemma 7.8 below implies that
0< ξα
M(h) = E[w(o)h(o)]
. This implies that
w
is not
identical to zero.
The above proof uses the following lemma.
Lemma 7.8. Let [D,o]be a unimodular discrete metric space.
(i) By letting b:= ξα
1(1), one has b≤ Hα
1(D)≤b+b|log b|.
(ii) Let h∈L1(D∗, µ)be a non-negative function. For M≥1, one has
Hα
M(D)≤inf
a≥0nMαEhe−ah(o)i+aξα
M(h)o.(7.5)
(iii) In addition, if h > 0a.s., then ξα
M(h)=0if and only if Hα
M(D) = 0.
Proof. (i)
. The first inequality is easily obtained from the definition of
ξα
1(1)
by consid-
ering the cases where
c(·)∈ {0,1}
. In particular, this implies that
b≤1
. The second
inequality is implied by part (ii) by letting h(·) := 1 and a:= −log b≥0.
(ii)
. Let
b0> ξα
M(h)
be arbitrary. So there exists an equivariant weighted collection of
balls
c
that satisfies
(7.1)
for
h
and
EhPr≥Mcr(o)rαi≤b0
. Next, given
a≥0
, define an
equivariant covering
R
as follows. For each
v∈D
and
r≥M
such that
cr(v)>0
, put a
ball of radius
r
at
v
with probability
acr(v)∧1
. Do this independently for all
v
and
r
(one
should condition on
D
first). If more than one ball is put at
v
, keep only the one with
maximum radius. Let
S
be the union of the chosen balls. For
u∈D\S
, put a ball of
EJP 26 (2021), paper 155. Page 49/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
radius
M
at
u
. This gives an equivariant covering, namely
R
, by balls of radii at least
M
.
Then, one gets
E[R(o)]α≤MαP[o6∈ S] + E
X
r≥M
(acr(o)∧1)rα
≤MαP[o6∈ S] + ab0.(7.6)
To bound
P[o6∈ S]
, consider a realization of
[D,o]
. First, if for some
v∈D
and
r≥M
, one has
acr(v)>1
and
o∈Nr(v)
, then
o
is definitely in
S
. Second, assume this
is not the case. By
(7.1)
, one has
Pu∈DPr≥Mcr(u)1{o∈Nr(u)}≥h(o)
. This implies that
the probability that o6∈ Sin this realization is
Y
(v,r):o∈Nr(v)
(1 −acr(v)) ≤exp 
−X
(v,r):o∈Nr(v)
acr(v)
≤e−ah(o).
In both cases, one gets
P[o6∈ S]≤Ee−ah(o)
. Thus,
(7.6)
implies that
E[R(o)]α≤
MαEe−ah(o)+ab0. Since a≥0and b0> b are arbitrary, the claim follows.
(iii)
. Assume
ξα
M(h)=0
. By letting
a→ ∞
in
(7.5)
and using dominated convergence,
one obtains that
Hα
M(D)=0
. Conversely, assume
Hα
M(D)=0
. The first inequality in
(i)
gives that
ξα
M(a)=0
for any constant
a
. Therefore,
ξα
M(h)≤ξα
M(a) + ξα
M((h−a)∨0)) ≤
MαE[(h−a)∨0]. By letting atend to infinity, one gets ξα
M(h)=0.
Remark 7.9.
In this subsection, the following metric is used on the mark space
Ξ
. Let
Ξ0
be the set of finite measures on
R2
. By identifying
c∈Ξ
with the counting measure
on the finite set
{(x, c(x)) : x∈R≥0, c(x)>0}
, one can identify
Ξ
with a Borel subset of
Ξ0
. It is well known that
Ξ0
is a complete separable metric space under the Prokhorov
metric (see e.g., [
17
]). So one can define the notion of
Ξ0
-valued equivariant processes
as in Definition 2.9. Therefore, Ξ-valued equivariant processes also make sense.
7.2 Max-Flow Min-Cut Theorem for Unimodular One-Ended Trees
The result of this subsection is used in the next subsection for a Euclidean version of
the unimodular Frostman lemma, but is of independent interest as well.
The max-flow min-cut theorem is a celebrated result in the field of graph theory
(see e.g., [
25
]). In its simple version, it studies the minimum number of edges in a
cut-set in a finite graph; i.e., a set of edges the deletion of which disconnects two given
subsets of the graph. A generalization of the theorem in the case of trees is obtained
by considering cut-sets separating a given finite subset from the set of ends of the tree.
This generalization is used to prove a version of Frostman’s lemma for compact sets in
the Euclidean space (see e.g., [14]).
This subsection presents an analogous result for unimodular one-ended trees. It
discusses cut-sets separating the set of leaves from the end of the tree. Since the tree
has infinitely many leaves a.s. (see e.g., [
5
]), infinitely many edges are needed in any
such cut-set. Therefore, cardinality cannot be used to study minimum cut-sets. The idea
is to use unimodularity for a quantification of the size of a cut-set.
Let
[T,o;c]
be a unimodular marked one-ended tree with mark space
R≥0
. Assume
the mark
c(e)
of each edge
e
is well defined and call it the
conductance
of
e
. Let
L
be
the set of leaves of
T
. As in Subsection 4.1.2, let
F(v)
be the parent of vertex
v
and
D(v)
be the descendants subtree of v.
Definition 7.10.
A
legal equivariant flow
on
[T;c]
is an equivariant way of assigning
extra marks
f(·)∈R
to the edges (see Definition 2.9 and Remark 2.13), such that almost
surely,
EJP 26 (2021), paper 155. Page 50/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
(i) for every edge e, one has 0≤f(e)≤c(e),
(ii) for every vertex v∈T\L, one has
f(v, F (v)) = X
w∈F−1(v)
f(w, v).(7.7)
Also, an
equivariant cut-set
is an equivariant subset
Π
of the edges of
[T;c]
that
separates the set of leaves Lfrom the end in T.
Note that extra randomness is allowed in the above definition. The reader can
think of the value
f(v, F (v))
as the flow from
v
to
F(v)
. So
(7.7)
can be interpreted as
conservation of flow at the vertices except the leaves. Also, the leaves are regarded as
the sources of the flow.
Since the number of leaves is infinite a.s., the sum of the flows exiting the leaves
might be infinite. In fact, it can be seen that unimodularity implies that the sum is
always infinite a.s. The idea is to use unimodularity to quantify how large is the flow.
Similarly, in any equivariant cut-set, the sum of the conductances of the edges is infinite
a.s. Unimodularity is also used to quantify the conductance of an equivariant cut-set.
These are done in Definition 7.12 below.
Below, since each edge of
T
can be uniquely represented as
(v, F (v))
, the following
convention is helpful.
Convention 7.11.
For the vertices
v
of
T
, the symbols
f(v)
and
c(v)
are used as
abbreviations for
f(v, F (v))
and
c(v, F (v))
, respectively. Also, by
v∈Π
, one means that
the edge (v, F (v)) is in Π.
Definition 7.12. The norm of the legal equivariant flow fis defined as
|f|:= Ef(o)1{o∈L}.
Also, for the equivariant cut-set Π, define
c(Π) := Ec(o)1{o∈Π}=E
X
w∈F−1(o)
c(w)1{w∈Π}
,
where the last equality follows from the mass transport principle (2.2).
An equivariant cut-set Πis called equivariantly minimal if there is no other equiv-
ariant cut-set which is a subset of
Π
a.s. If so, it can be seen that it is
almost surely
minimal
as well; i.e., in almost every realization, it is a minimal cut set (see Lemma C.6
of [7]).
Lemma 7.13.
If
f
is a legal equivariant flow and
Π
is an equivariant cut-set, then
|f| ≤ c(Π). Moreover, if the pair (f,Π) is equivariant, then
|f| ≤ Ef(o)1{o∈Π}≤c(Π).
In addition, if Πis minimal, then equality holds in the left inequality.
Proof. One can always consider an independent coupling of fand Π(as in the proof of
Theorem 5.2). So assume
(f,Π)
is equivariant from the beginning. Note that the whole
construction (with conductances, the flow and the cut-set) is unimodular (Lemma 2.12).
For every leaf
v∈L
, let
τ(v)
be the first ancestor of
v
such that
(v, F (v)) ∈Π
. Then,
EJP 26 (2021), paper 155. Page 51/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
send mass
f(v)
from each leaf
v
to
τ(v)
. By the mass transport principle (2.2), one gets
Ef(o)1{o∈L}=E
1{o∈Π}X
v∈τ−1(o)
f(v)

≤E
1{o∈Π}X
v∈D(o)∩L
f(v)
=Ef(o)1{o∈Π},
where the last equality holds because
f
is a flow. Moreover, if
Π
is minimal, then the
above inequality becomes an equality and the claim follows.
The main result is the following converse to the above lemma.
Theorem 7.14 (Max-Flow Min-Cut for Unimodular One-Ended Trees).For every unimod-
ular marked one-ended tree
[T,o;c]
equipped with conductances
c
as above, if
c
is
bounded on the set of leaves, then
max
f|f|= inf
Πc(Π),
where the maximum is over all legal equivariant flows
f
and the infimum is over all
equivariant cut-sets Π.
Remark 7.15.
The claim of Theorem 7.14 is still valid if the probability measure (the
distribution of
[T,o;c]
) is replaced by any (possibly infinite) measure
P
on
D0
∗
supported
on one-ended trees, such that
P(o∈L)<∞
and the mass transport principle (2.2)
holds. The same proof works for this case as well. This will be used in Subsection 7.3.
Proof of Theorem 7.14.
For
n≥1
, let
Tn
be the sub-forest of
T
obtained by keeping
only vertices of height at most
n
in
T
. Each connected component of
Tn
is a finite tree
which contains some leaves of
T
. For each such component, namely
T0
, do the following:
if
T0
has more than one vertex, consider the maximum flow on
T0
between the leaves
and the top vertex (i.e., the vertex with maximum height in
T0
). If there is more than one
maximum flow, choose one of them randomly and uniformly. Also, choose a minimum
cut-set in
T0
randomly and uniformly. Similarly, if
T0
has a single vertex
v
, do the same
for the subgraph with vertex set
{v, F (v)}
and the single edge adjacent to
v
. By doing
this for all components of
Tn
, a (random) function
fn
on the edges and a cut-set
Π0
n
are
obtained (by letting
fn
be zero on the other edges).
Π0
n
is always a cut-set, but
fn
is not
a flow. However,
fn
satisfies (7.7) for vertices of
Tn\L
, except the top vertices of the
connected components of Tn. Also, it can be seen that fnand Π0
nare equivariant.
For each component
T0
of
Tn
, the set of leaves of
T0
, excluding the top vertex, is
L∩T0
. So the max-flow min-cut theorem of Ford-Fulkerson [
25
] (see e.g., Theorem 3.1.5
of [14]) gives that, for each component T0of Tn, one has
X
v∈L∩T0
fn(v) = X
e∈Π0
n∩T0
c(e).
If
u
is the top vertex of
T0
, let
h(u)
be the common value in the above equation. By using
the mass transport principle (2.2) for each of the two representations of
E[h(o)]
, one
can obtain
Efn(o)1{o∈L}=E[h(o)] =Ec(o)1{o∈Π0
n}=c(Π0
n).
Since
0≤fn(·)≤cn(·)
, one can see that the distributions of
fn
are tight (see [
7
]).
Therefore, there is a sequence
n1, n2, . . .
and an equivariant process
f0
such that
fni→f0
(weakly). It is not hard to deduce that
f0
is a legal equivariant flow. Also, since
f0(o)
EJP 26 (2021), paper 155. Page 52/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
and
1{o∈L}
are continuous functions of
[T,o;f0]
and their product is bounded (by the
assumption on c), one gets that
f0=Ef0(o)1{o∈L}= lim
i
Efni(o)1{o∈L}= lim
ic(Π0
ni).
Therefore,
maxf|f| ≥ infΠc(Π)
. Note that the maximum of
|f|
is attained by the same
tightness argument as above. So Lemma 7.13 implies that equality holds and the claim
is proved.
7.3 A Unimodular Frostman Lemma for Point Processes
In the Euclidean case, another form of the unimodular Frostman lemma is given
below. Its proof is based on the max-flow min-cut theorem of Subsection 7.2. As will be
seen, the claim implies that in this case, Conjecture 7.4 holds up to a constant factor
(Corollary 7.17). However, the weight function obtained in the theorem needs extra
randomness.
Theorem 7.16.
Let
Φ
be a point-stationary point process in
Rk
endowed with the
l∞
metric, and let
α≥0
. Then, there exists an equivariant weight function
w
on
Φ
such
that, almost surely,
∀v∈Φ,∀r≥1 : w(Nr(v)) ≤rα(7.8)
and
E[w(0)] ≥3−kHα
1(Φ).(7.9)
In particular, if Hα
1(Φ) >0, then w(0) is not identical to zero.
A similar result holds for the Euclidean metric or other equivalent metrics by just
changing the constant 3−kin (7.9).
In the following proof,
Φ
is regarded as a counting measures; i.e., for all
A⊆Rd
,
Φ(A) := #(Φ ∩A).
Proof.
Let
b > 1
be an arbitrary integer (e.g.,
b= 2
). x For every integer
n≥0
, let
Qn
be the stationary partition of
Rk
by translations of the cube
[0, bn)k
as in Subsection 5.3.
Consider the nested coupling of these partitions for
n≥0
(i.e., every cube of
Qn
is
contained in some cube of
Qn+1
for every
n≥0
) independent of
Φ
. Let
T0
be the tree
whose vertices are the cubes in
∪nQn
and the edges are between all pairs of nested
cubes in
Qn
and
Qn+1
for all
n
. Let
T⊆T0
be the subtree consisting of the cubes
qn(v)
for all
v∈Φ
and
n≥0
. The set
L
of the leaves of
T
consists of the cubes
q0(v)
for
all
v∈Φ
. Let
σ:= q0(0) ∈L
. Note that in the correspondence
v7→ q0(v)
, each cube
σ∈L
corresponds to
Φ(σ)≥1
points of
Φ
. Therefore, by verifying the mass transport
principle, it can be seen that the distribution of
[L,σ]
, biased by
1/Φ(σ)
, is unimodular;
i.e.,
E"1
Φ(σ)X
σ0∈L
g(L,σ, σ0)#=E"1
Φ(σ)X
σ0∈L
g(L, σ0,σ)#,
for every measurable
g≥0
. In addition,
g
can be allowed to depend on
T
in this equation
(but the sum is still on
σ0∈L
). Therefore, one can assume the metric on
L
is the graph-
distance metric induced from
T
(see Theorem 3.31). Moreover, Theorem 5 of [
35
] implies
that by a further biasing and choosing a new root for
T
, one can make
T
unimodular.
More precisely, the following (possibly infinite) measure on D∗is unimodular:
P[A] := E
X
n≥0
1
en
1A[T,qn(0)]
,(7.10)
EJP 26 (2021), paper 155. Page 53/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
where
en:= Φ(qn(0))
. Let
E
denote the integral operator w.r.t. the measure
P
. For any
equivariant flow
f
on
T
, the norm of
f
w.r.t. the measure
P
(see Remark 7.15) satisfies
|f|=E[f·1L] = E
X
n≥0
1
en
f(qn(0))1{qn(0)∈L}
=E1
Φ(σ)f(σ),
where the second equality is by
(7.10)
. Consider the conductance function
c(τ) := bnα
for
all cubes
τ
of edge length
bn
in
T
and all
n
. Therefore, Theorem 7.14 and Remark 7.15
imply that the maximum of
E[f(σ)]
over all equivariant legal flows
f
on
[T,σ]
is attained
(note that
[T,σ]
is not unimodular, but the theorem can be used for
P
). Denote by
f0
the
maximum flow. Let
w
be the weight function on
Φ
defined by
w(v) = δf0(q0(v))/Φ(q0(v))
,
for all
v∈Φ
, where
δ:= (b+ 1)−k
. The claim is that
w
satisfies the requirements
(7.8)
and (7.9). Since f0is a legal flow, it follows that for every cube σ∈T, one has
w(σ) = δf0(σ)≤δc(σ) = δbnα .
Each cube
σ
of edge length
r∈[bn, bn+1)
in
Rk
can be covered with at most
(b+ 1)k
cubes of edge length
bn
in
T0
. If
n≥0
, the latter are either in
T
or do not intersect
Φ
.
So the above inequality implies that w(σ)≤rα. So (7.8) is proved for w.
To prove
(7.9)
, given any equivariant cut-set
Π
of
T
, a covering of
Φ
can be con-
structed as follows: For each cube
σ∈Π
of edge length say
bn
, let
τ(σ)
be one of the
points in
σ∩Φ
chosen uniformly at random and put a ball of radius
bn
centered at
τ(σ)
.
Note that this ball contains
σ
. Do this independently for all cubes in
T
. If a point in
Φ
is
chosen more than once, consider only the largest radius assigned to it. It can be seen
that this gives an equivariant covering of Φ, namely R. One has
E[R(0)α]≤E
X
n≥0
bnα1{qn(0)∈Π}1{0=τ(qn(0))}

=E
X
n≥0
bnα
en
1{qn(0)∈Π}
.
On the other hand, by (7.10), one can see that
c(Π) = E
X
n≥0
1
en
c(qn(0))1{qn(0)∈Π}
=E
X
n≥0
bnα
en
1{qn(0)∈Π}
.
Therefore,
E[R(0)α]≤c(Π)
. So
Hα
1(Φ) ≤c(Π)
. Since
Π
is an arbitrary equivariant
cut-set, by the unimodular max-flow min-cut theorem established above (Theorem 7.14)
and the maximality of the flow
f0
, one gets that
Hα
1(Φ) ≤ |f0|=E[f0(σ)/Φ(σ)] =
δ−1E[w(0)]. So the claim is proved.
The following corollary shows that in the setting of Theorem 7.16, the claim of
Conjecture 7.4 holds up to a constant factor (compare this with Lemma 7.8).
Corollary 7.17.
For all point-stationary point processes
Φ
in
Rk
endowed with the
l∞
metric and all α≥0,3−kHα
1(Φ) ≤ξα
1(Φ) ≤ Hα
1(Φ).
Proof. The claim is directly implied by (7.2), Theorem 7.16 and Remark 7.3.
7.4 Applications
The following subsections give some basic applications of the unimodular Frostman
lemma. This lemma is also the basis of many results of [6].
EJP 26 (2021), paper 155. Page 54/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
7.4.1 Cayley Graphs
Proposition 7.18.
For every finitely generated group
H
with polynomial growth degree
α∈[0,∞], one has ξα
∞(H)<∞.
Note that if Conjecture 7.4 holds, then this result implies
Mα(H)>0
, as conjectured
in Subsection 6.6.
Proof.
By Theorem 6.14,
Mα(H)<∞
. So the unimodular Frostman lemma (Theo-
rem 7.2) implies that for every
M≥1
, there exists
w:D∗→R≥0
such that
w(NM(e)) ≤
Mα
and
E[w(e)] = ξα
M(H)
, where
e
is the neutral element of
H
. Since the Cayley
graph of
H
is transitive and
w
is defined up to rooted isomorphisms,
w(H, ·)
is con-
stant. Hence,
w(H, v) = ξα
M(H)
for all
v∈H
. Therefore,
ξα
M(H)#NM(e)≤Mα
. Thus,
ξα
M(H)≤1/c
, where
c
is as in the proof of Theorem 6.14. By letting
M→ ∞
, one gets
ξα
∞(H)≤1/c < ∞.
7.4.2 Dimension of Product Spaces
Let
[D1,o1]
and
[D2,o2]
be independent unimodular discrete metric spaces. By consid-
ering any of the usual product metrics; e.g., the sup metric or the
p
product metric, the
independent product [D1×D2,(o1,o2)]
makes sense as a random pointed discrete
space. It is not hard to see that the latter is also unimodular (see also Proposition 4.11
of [2]).
Proposition 7.19.
Let
[D1×D2,(o1,o2)]
represent the independent product of
[D1,o1]
and [D2,o2]defined above. Then,
udimH(D1) + udimM(D2)≤udimH(D1×D2)≤udimH(D1) + udimH(D2).(7.11)
Proof.
By Theorem 3.31, one can assume the metric on
D1×D2
is the sup metric
without loss of generality. So Nr(v1, v2) = Nr(v1)×Nr(v2).
The upper bound is proved first. For
i= 1,2
, let
αi>udimH(Di)
be arbitrary. By the
unimodular Frostman lemma (Theorem 7.2), there is a nonnegative measurable functions
wi
on
D∗
such that
∀v∈Di:∀r≥1 : wi(Nr(v)) ≤rα, a.s.
In addition,
wi(oi)6= 0
with
positive probability. Consider the equivariant weight function
w
on
D1×D2
defined
by
w(v1, v2) := w1[D1, v1]×w2[D2, v2]
. It is left to the reader to show that
w
is an
equivariant weight function. One has
w(Nr(v1, v2)) = w1(Nr(v1))w2(Nr(v2)) ≤rα1+α2
.
Also, by the independence assumption,
w(o1,o2)6= 0
with positive probability. Therefore,
the mass distribution principle (Theorem 5.2) implies that
udimH(D1×D2)≤α1+α2
.
This proves the upper bound.
For the lower bound in the claim, let
α < udimH(D1)
,
β < udimM(D2)
and
 > 0
be arbitrary. It is enough to find an equivariant covering
R
of
D1×D2
such that
ER(o1,o2)α+β< 
. One has
decay (λr(D2)) > β
, where
λr
is defined in (3.1). So there
is
M > 0
such that
∀r≥M:λr(D2)< r−β
. So for every
r≥M
, there is an equivariant
r
-covering of
D2
with intensity less than
r−β
. On the other hand, since
α < udimH(D1)
,
one has
Hα
M(D1) = 0
(by Lemma 3.25). Therefore there is an equivariant covering
R1
of
D1
such that
ER1(o1)β< 
and
∀v∈D1:R1(v)∈ {0} ∪ [M, ∞)
a.s. Choose the extra
randomness in
R1
independently from
[D2,o2]
. Given a realization of
[D1,o1]
and
R1
,
do the following: Let
v1∈D1
such that
R1(v1)6= 0
(and hence,
R1(v1)≥M
). One can
find an equivariant subset
Sv1
of
D2
that gives a covering of
D2
by balls of radius
R1(v1)
and has intensity less than
R1(v1)−β
. Do this independently for all
v1∈D1
. Now, for all
(v1, v2)∈D1×D2, define
R(v1, v2) := (R1(v1)if R1(v1)6= 0 and v2∈Sv1,
0otherwise.
EJP 26 (2021), paper 155. Page 55/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
Now,
R
is a covering of
D1×D2
and it can be seen that it is an equivariant covering.
Also, given
[D1,o1]
and
R1
, the probability that
o2∈So1
is less than
R1(o1)−β
. So one
gets
ER(o1,o2)α+β=E"EhR(o1,o2)α+β|[D1,o1],R1i#
<ER1(o1)α+βR1(o1)−β=E[R1(o1)α]< .
So the claim is proved.
The following examples provide instances where the inequalities in (7.11) are strict.
Example 7.20.
Assume
[D1,o1]
and
[D2,o2]
are unimodular discrete spaces such that
udimM(G1)<udimH(G1)and udimM(G2) = udimH(G2). By Proposition 7.19, one gets
udimH(G1×G2)≥udimH(G1) + udimM(G2)>udimH(G2) + udimM(G1).
So by swapping the roles of the two spaces, an example of strict inequality in the left
hand side of (7.11) is obtained.
Example 7.21.
Let
J
be a subset of
Z≥0
such that
d(J) = 1
and
d(J) = 0
simultaneously
(see Subsection 6.5.1 for the definitions). Let
Ψ1
and
Ψ2
be defined as in Subsection 6.5.1
corresponding to
J
and
Z≥0\J
respectively. Proposition 6.11 implies that
udimH(Ψ1) =
udimH(Ψ2) = 1. On the other hand, (6.2) implies that
#N2n(o1×o2)≤2Jn+1 ×2(n+1−Jn)+1 = 2n+3.
This implies that
growth (Nr(o)) ≤1
. So the unimodular Billingsley lemma (Theorem 5.6)
implies that
udimH(Ψ1×Ψ2)≤1
(in fact, equality holds by Proposition 7.24 below). So
the rightmost inequality in (7.11) is strict here.
7.4.3 Dimension of Embedded Spaces
It is natural to think of
Z
as a subset of
Z2
. However,
[Z,0]
is not an equivariant
subspace of
[Z2,0]
. By the following definition,
[Z,0]
is called embeddable in
[Z2,0]
. The
dimension of embedded subspaces is studied in this subsection. The analysis requires
the unimodular Frostman lemma.
Definition 7.22.
Let
[D0,o0]
and
[D,o]
be random pointed discrete spaces. An
em-
bedding
of
[D0,o0]
in
[D,o]
is a (not necessarily unimodular) random pointed marked
discrete space [D0,o0;m]with mark space {0,1}such that
(i) [D0,o0]has the same distribution as [D,o].
(ii) m(o0) = 1
a.s. and by letting
S:= {v∈D0:m(v) = 1}
equipped with the induced
metric from D0,[S,o0]has the same distribution as [D0,o0].
If in addition,
[D0,o0]
is unimodular, then
[D0,o0;m]
is called an
equivariant embed-
ding if
(iii)
The mass transport principle holds on
S
; i.e., (2.2) holds for functions
g(u, v) :=
g(D0, u, v;m)such that g(u, v)is zero when m(u)=0or m(v) = 0.
If an embedding (resp. an equivariant embedding) exists,
[D0,o0]
is called
embeddable
(resp. equivariantly embeddable) in [D,o].
It should be noted that
[D0,o0;m]
is not an equivariant process on
D
except in the
trivial case where m(·) = 1 a.s.
EJP 26 (2021), paper 155. Page 56/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
Example 7.23. Here are instances of Definition 7.22.
(i)
Let
[D0,o0] := [Z,0]
and
[D,o] := [Z2,0]
equipped with the sup metric. Consider
m:Z2→ {0,1}
which is equal to one on the boundary of the positive cone. Then,
[Z2,0; m]
is an embedding of
[Z,0]
in
[Z2,0]
, but is not an equivariant embedding
since it does not satisfy (iii).
(ii)
A point-stationary point process in
Zk
(pointed at 0) is equivariantly embeddable in
[Zk,0].
(iii)
Let
H
be a finitely generated group equipped with the graph-distance metric of
an arbitrary Cayley graph over
H
. Then, any subgroup of
H
(equipped with the
induced metric) is equivariantly embeddable in H.
Proposition 7.24.
If
[D0,o0]
and
[D,o]
are unimodular discrete spaces and the former
is equivariantly embeddable in the latter, then
udimH(D)≥udimH(D0),(7.12)
and for all α≥0and M≥1, with ξα
Mdefined in Definition 7.1,
ξα
M(D,1) ≤ Hα
M(D0).(7.13)
Proof.
First, assume
(7.13)
holds. For
α > udimH(D)
, one has
Hα
M(D)>0
(Lemma 3.25).
Therefore, Lemma 7.8 implies that
ξα
M(D,1) >0
. Hence,
(7.13)
implies that
Hα
M(D0)>0
,
which implies that udimH(D0)≤α. So it is enough to prove (7.13).
By the unimodular Frostman lemma (Theorem 7.2), there is a bounded function
w:D∗→R≥0
such that
E[w(o)] = ξα
M(D,1)
, and almost surely,
w(Nr(o)) ≤rα
for all
r≥M
. Assume
[D0,o0;m]
is an equivariant embedding as in Definition 7.22. For
x∈D0
,
let
w0(x) := w0
D0(x) := w[D0, x]
. Consider the random pointed marked discrete space
[S,o0;w0]
obtained by restricting
w0
to
S
. By the definition of equivariant embeddings and
by directly verifying the mass transport principle, the reader can obtain that
[S,o0;w0]
is
unimodular. Since
[S,o0]
has the same distribution as
[D0,o0]
, there exists an equivariant
process
w0
on
D0
such that
[S,o0;w0]
has the same distribution as
[D0,o0;w0]
(see the
converse of Lemma 2.12 in Subsection 2.5). According to the above discussion, one has
∀r≥M:w0(Nr(S,o0)) ≤w0(Nr(D0,o0)) ≤rα, a.s.
This implies that
w0(Nr(o0)) ≤rα
a.s. Therefore, the mass distribution principle (Theo-
rem 5.2) implies that E[w0(o0)] ≤ Hα
M(D0). One the other hand,
E[w0(o0)] = E[w0(o0)] = E[w(o)] = ξα
M(D,1),
where the last equality is by the assumption on
w
. This implies that
Hα
M(D0)≥ξα
M(D,1)
and the claim is proved.
It is natural to expect that an embedded space has a smaller Hausdorff size. This is
stated in Conjecture 8.4.
Remark 7.25.
Another possible way to prove Proposition 7.24 and Conjecture 8.4 is to
consider an arbitrary equivariant covering of
D0
and try to extend it to an equivariant
covering of
D
by adding some balls (without adding a ball centered at the root). More
generally, given an equivariant processes
Z0
on
D0
, one might try to extend it to an
equivariant process on
D
without changing the mark of the root. But at least the latter
is not always possible. A counter example is when
[D0,o0]
is
K2
(the complete graph
with two vertices),
[D,o]
is
K3
,
Z0(o0) = ±1
chosen uniformly at random, and the mark
of the other vertex of D0is −Z0(o0).
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Unimodular Hausdorff and Minkowski dimensions
7.5 Notes and Bibliographical Comments
The unimodular Frostman lemma (Theorem 7.2) is analogous to Frostman’s lemma
in the continuum setting (see e.g., Thm 8.17 of [
42
]). The proof of Theorem 7.2 is also
inspired by that of [
42
], but there are substantial differences. For instance, the proof of
Lemma 7.8 and also the use of the duality of
L1
and
L∞
in the proof of Theorem 7.2 are
new. The Euclidean version of the unimodular Frostman lemma (Theorem 7.16) and its
proof are inspired by the continuum analogue (see e.g., [14]).
As already explained, the unimodular max-flow min-cut theorem (Theorem 7.14) is
inspired by the max-flow min-cut theorem for finite trees. Also, the results and examples
of Subsection 7.4.2 on product spaces are inspired by analogous in the continuum setting;
e.g., Theorem 3.2.1 of [14].
8 Miscellaneous Topics
8.1 Connections with Other Notions of Dimension
Several notions of dimension are already defined in the literature for discrete spaces
in special cases. A few of them are listed in this subsection together with their connec-
tions to unimodular dimensions.
For subsets of
Zd
, the notions of upper and lower mass dimension are defined in [
8
],
which are just the volume growth rates defined in Section 5. The paper [
38
] extends the
upper mass dimension to general subsets
A⊆Rd
and calls it the macroscopic Minkowski
dimension of
A
(one may define lower macroscopic Minkowski dimension similarly). This
extension is obtained by pixelizing
A
to get a subset of
Zd
. The unimodular Billingsley
lemma states that for unimodular (i.e., point-stationary) and ergodic subsets of
Zd
, the
unimodular Hausdorff dimension is between the upper and lower mass dimension. A
similar result holds in the non-integer case as well:
Corollary 8.1.
For ergodic point-stationary point processes in
Rd
, the unimodular
Hausdorff dimension is between the upper and lower macroscopic Minkowski dimensions
a.s.
This is a direct corollary of Billingsley’s lemma applied to the pixelization by a
randomly-shifted lattice. It can also be proved by using weights in Billingsley’s lemma
similarly to the proof of Proposition 5.15.
Another notion is that of discrete (Hausdorff) dimension [
8
], which uses the idea
behind the definition of the classical Hausdorff dimension by considering coverings of
Φ⊆Zd
by large balls and considering the cost
(r
r+|x|)α
for each ball in the covering,
where
r
and
x
are the radius and the center of the ball and
α
is a constant (in fact, this
is a modified version of the definition of [
8
] mentioned in [
14
]). In the future work [
6
], it
is shown that the discrete dimension is an upper bound for the unimodular Hausdorff
dimension, when both notions are defined (i.e., for point-stationary point processes).
The unimodular Hausdorff dimension can be connected to the classical Hausdorff
dimension via scaling limits. Such limits are random continuum metric spaces and can
be defined by weak convergence w.r.t. the Gromov-Hausdorff-Prokhorov metric [
37
]. It
is shown in the preprint [
6
] that if the unimodular discrete space admits a scaling limit,
then the ordinary Hausdorff dimension of the limit is an upper bound for the unimodular
Hausdorff dimension.
The above inequalities are expected to be equalities in most examples. The preprint
[
6
] provides more discussion on the matter. Note that these comparison results imply
relations between the volume growth rate, scaling limits and discrete dimension, which
are of independent interest and which are new to the best of the authors’ knowledge.
A problem of potential interest is the connection of unimodular dimensions to other
EJP 26 (2021), paper 155. Page 58/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
notions of dimension. This includes Gromov’s notion of asymptotic dimension [
28
],
the spectral dimension of a graph (defined in terms of the return probabilities of the
simple random walk), the typical displacement exponent of a graph (see [
16
] for both
notions), the isoperimetric dimension of a graph [
15
], the resistance growth exponent
of a graph, the stochastic dimension of a partition of
Zd
[
11
], etc. In statistical physics,
one also assigns dimension and various exponents to finite models. Famous examples
are self-avoiding walks and the boundaries of large percolation clusters.
8.2 Gauge Functions and the Unimodular Dimension Function
There exist unimodular discrete spaces
D
in which the
udimH(D)
-dimensional Haus-
dorff size is either zero or infinity (e.g., Examples 6.1 and 6.2). For such spaces, it is
convenient to generalize the unimodular Hausdorff size as follows. Consider an increas-
ing function
ϕ:{0} ∪ [1,∞)→[0,∞)
; e.g.,
ϕ(r) = rα
, called a gauge function. Define
Hϕ
M(D)
by
infR{E[ϕ(R(o))]}
similarly to
(3.3)
. Then, define
Mϕ(D)
similarly to
(3.6)
.
If 0<Mϕ(D)<∞, then ϕis called a unimodular dimension function for D.
In addition, given a family of gauge functions
(ϕα)α≥0
that is increasing in
α
and
such that
∀α > β : limr→∞ ϕα(r)/ϕβ(r) = ∞
, one can redefine the unimodular Hausdorff
dimension by
sup{α:Mϕα(D)=0}
(see e.g., the next paragraph). One can redefine the
unimodular Minkowski dimension similarly. The authors have verified that the results
of the paper can be extended to this setting except that Theorem 3.34 and the results
of Subsection 3.8 require the doubling condition
supr≥1ϕ(2r)/ϕ(r)<∞
. The general
result of Subsection 4.1.2 can also be extended under the doubling condition. Also, the
upper bounds in the unimodular mass distribution principle, the unimodular Billingsley
lemma and the unimodular Frostman lemma hold in this more general setting (some
other results require the doubling condition). However, for the ease of reading, the
results are presented in the original setting of this paper.
As an example of the above framework, one can define the
exponential dimension
by considering
ϕα(r) := eαr
. It might be useful for studying unimodular spaces with
super-polynomial volume growth, which are more interesting in group theory (see
Subsection 6.6). Other gauge functions may also be useful for groups of intermediate
growth. Note that exponential gauge functions do not satisfy the doubling condition, and
hence, the reader should be careful about using the results of this work for such gauge
functions.
8.3 Negative Dimensions
If a compact metric space
X
is the union of
k
disjoint copies of
1
rX
, then the similarity
dimension of
X
is
log k/ log r
(see e.g., [
14
]). This definition can also be used for some
infinite discrete sets as well. For instance,
Zd
is a union of
2d
copies of
2Zd
. So, it can
be said that the similarity dimension of
Zd
is negative. The (deterministic) discrete
Cantor set (see e.g., [
14
]) is also
(−log 2/log 3)
-dimensional. There are several further
arguments, listed below, suggesting that one should actually assign negative dimensions
to unimodular discrete spaces.
First, this would be natural in terms of definition. The unimodular Minkowski di-
mensions should be redefined by
udimM(D) = growth (λr)
and
udimM(D) = growth (λr)
.
Using growth instead of decay would then unify the definition of the ordinary Minkowski
dimension of compact sets and the unimodular Minkowski dimension. The former is
microscopic (i.e., when
r
tends to 0), whereas the latter is macroscopic (
r→ ∞
). One
may also replace the unimodular Hausdorff by the negative of the definitions given so
far.
Secondly, this unification of the definitions would also take care of the puzzling
EJP 26 (2021), paper 155. Page 59/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
direction of certain inequalities discussed in the paper: when adopting these negative
unimodular dimensions, (i) the classical and unimodular Minkowski and Hausdorff di-
mensions would be ordered in the same way, i.e.,
udimH(D)≤udimM(D)≤udimM(D)
;
(ii) an equivariant subset of a unimodular set would have a unimodular Minkowski
dimension smaller than or equal to that of the set, and possibly strictly smaller (see Sub-
sections 3.7 and 4.4), a property that is expected to hold for any notion of dimension; (iii)
the mass distribution principle and the Billingsley lemma would provide lower bounds on
udimH(D)
, while upper bounds would be obtained by constructing explicit coverings; (iv)
the dimension of non-ergodic examples (e.g., Example 3.19) would also be the supremum
dimension of the components, as one might expect (see also Remark 3.20).
It should however be noted that by assigning such negative dimension, the dimension
of a non-equivariant subset (see Subsection 7.4.3) would be larger than or equal to that
of the whole space (just as the similarity dimension of
Z
is larger than that of
Z2
). One
should not expect that non-equivariant subsets behave like equivariant subsets, since
they do not satisfy the main assumption of statistical homogeneity, which is the basis of
all of the definitions in this work.
8.4 Problems and Conjectures
This subsection gathers some problems and conjectures pertaining to the theory of
unimodular dimensions and to specific examples. Those are already stated in the paper
and are also briefly listed.
8.4.1 Further conjectures and problems
1. Connections to other notions
. The following conjectures connect unimodular
dimensions to the properties of the simple random walk.
Conjecture 8.2.
If
[G,o]
is a unimodular graph such that
udimH(G)<2
, then the simple
random walk in Gis recurrent a.s.
Note that the converse of this conjecture does not hold. For instance, it is not hard to
show that any one-ended tree (e.g., the canopy tree) is recurrent.
In [
8
], the discrete dimension of
A⊆Zd
is connected to whether the simple random
walk in
Zd
hits
A
infinitely often or not. Analogously, one can generalized the above
conjecture as follows:
Conjecture 8.3.
If
[D,o]
is equivariantly embedded in the unimodular graph
[G,o]
and
udimH(D)>udimH(G)−2
, then the simple random walk in
G
hits
D
infinitely often a.s.
2. Dimension functions.
Does there exist a unimodular discrete space without any
dimension function? The answer is not known yet. [
22
] gives a positive answer to the
analogous question in the continuum setting, but the proof ideas don’t seem to work
in the unimodular discrete setting. Also, by analogy with stable trees [
19
], a possible
candidate is unimodular EGW trees with infinite offspring variance.
3. Simple random walk.
By analogy with the image of subordinator processes (see
e.g., [
26
]), one may guess the exact unimodular dimension function for the image of
the random walk under the assumptions of Proposition 4.9. For example, consider the
zero set
Ψ
of the simple random walk (Proposition 4.10). By analogy with the zero set
of Brownian motion [
48
], it is natural to guess that
M1/2(Ψ) = ∞
and
√rlog log r
is a
dimension function for
Ψ
. To prove this, one should strengthen the bound
Crβlog log r
in the proof of Proposition 4.9 and also construct a covering of the set which is better
than that of Proposition 3.13. For the former, one may use Theorem 4 of [
26
] (it seems
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Unimodular Hausdorff and Minkowski dimensions
that the assumption of [
26
] on the tail of the jumps is not necessary for having only an
upper bound). For the latter, one might try to get ideas from [
48
] (it is necessary to use
intervals with different lengths).
Another guess is that the image of the symmetric nearest-neighbor simple random walk
in
Zd
is 2-dimensional when
d≥3
. More generally, if the jumps are in the domain of
attraction of a symmetric
α
-stable process, then the image is
α
-dimensional. These might
be proved similarly to the analogous results in [9].
For the graph of the simple random walk equipped with the Euclidean metric (Subsec-
tion 6.4), the guess is that if the increments are in the domain of attraction of an
α
-stable
distribution, where
0< α ≤2
, then
udimM(Ψ) = udimH(Ψ) = min{1,max(0,2α−1)/α}
(see Theorem 3.13 of [
38
]). Also, the guess is that the zero set of the symmetric
nearest-neighbor simple random walk in Z2is 1
4-dimensional.
4. Eternal Galton-Watson trees.
For unimodular eternal Galton-Watson trees (Sub-
section 4.2.3), a conjecture is that if the offspring distribution is in the domain of at-
traction of an
α
-stable distribution, where
α∈[1,2]
, then
udimM(T) = udimH(T) = α
α−1
(see [
29
] or Theorem 5.5 of [
20
]). The guess is that there is no regularly varying dimen-
sion function (see [
19
]), except in the finite-variance case (
α= 2
), where one may guess
that the dimension function is r2log log r(see [21]).
5. Drainage networks.
One can ask about the dimension of other drainage network
models. In particular, the simple model of Subsection 4.5 can be extended to a model in
Zkfor k > 2and the connected component containing the origin is unimodular.
6. Embedded spaces.
Conjecture 8.4.
Under the setting of Proposition 7.24, for all
α > 0
, one has
Mα(D)≥
Mα(D0).
Note that in the case
α= 0
, the conjecture is implied by Proposition 3.28. Also, in the
general case, the conjecture is implied by
(7.13)
and Conjecture 7.4. Another problem is
the validity of Proposition 7.24 under the weaker assumption of being non-equivariantly
embeddable. As a partial answer, if
growth (#Nr(o))
exists, then
(7.12)
holds. This is
proved as follows:
udimH(D0)≤ess inf growth (#Nr(o0)) ≤ess inf growth (#Nr(o))
= ess inf growth (#Nr(o)) = udimH(D),
where the first inequality and the last equality are implied by the unimodular Billingsley
lemma.
8.4.2 List of conjectures and problems mentioned in the previous sections
It is not known whether the lower bound
(4.3)
for the Hausdorff dimension of unimodular
one-ended trees is always an equality or not. Problem 5.12 asks whether the equality
udimH(D) = growth (w(Nr(o)))
always holds. This is implied by Problem 5.21, which
states that the upper and lower growth rates of
w(Nr(o))
(used in Billingsley’s lemma)
do not depend on w.
Conjecture 5.17 states that every point-stationary point process has zero Hausdorff
size unless when it is the Palm version of some stationary point process. Conjecture 7.4
states that
Hα
M(D) = ξα
M(1)
, where
ξα
M(1)
is define in Subsection 7.1. This implies the
conjecture that Cayley graphs have positive Hausdorff size (see Subsection 6.6).
EJP 26 (2021), paper 155. Page 61/64 https://www.imstat.org/ejp

Unimodular Hausdorff and Minkowski dimensions
It would be interesting to find connections between unimodular dimensions and other
notions of dimension, some of which are discussed in Subsection 8.1. Also, as mentioned
in the introduction and Subsection 8.2, the setting of this paper might be useful in the
study of examples pertaining to statistical physics or group theory.
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