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Unimodular Hausdorff and Minkowski
Dimensions
Fran¸cois Baccelli∗
, Mir-Omid Haji-Mirsadeghi†
, and Ali Khezeli‡
February 14, 2021
Abstract
This work introduces two new notions of dimension, namely the uni-
modular Minkowski and Hausdorff dimensions, which are inspired from
the classical analogous notions. These dimensions are defined for unimod-
ular 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 def-
initions of dimension is novel. Also, a toolbox of results is presented
for the analysis of these dimensions. In particular, analogues of Billings-
ley’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 uni-
modular one-ended trees and a weak form of pointwise ergodic theorems
for all unimodular discrete spaces.
1 Introduction
Infinite discrete random structures are ubiquitous: random graphs, branching
processes, point processes, graphs or zeros of discrete random walks, discrete or
continuum percolation, to name a few. The large scale and macroscopic proper-
ties of such spaces have been thoroughly discussed in the literature. In particu-
lar, various notions of dimension have been proposed; e.g., the mass dimension
∗The University of Texas at Austin and INRIA Paris, baccelli@math.utexas.edu
†Sharif University of Technology, mirsadeghi@sharif.ir
‡Tarbiat Modares University, khezeli@modares.ac.ir, current address: INRIA Paris,
ali.khezeli@inria.fr
1
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 homogeneity. 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 uni-
fies 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 de-
fine 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 clas-
sical 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 unimod-
ular dimensions. 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 state-
ments 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 no-
tions 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 di-
mensions. 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 unimod-
ular 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
2
between the volume growth rate and the height of the root is established as
explained below.
The framework is used to derive concrete results on the dimension of sev-
eral instances of unimodular random discrete metric spaces. This is done for
the zeros and the graph of discrete random walks, sets defined by digit restric-
tion, 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 unimod-
ular one-ended tree and the tail of the distribution 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 mathe-
matical physics. Many examples in this domain enjoy some kind of homogeneity
and give rise to unimodular spaces, directly or indirectly; e.g., percolation clus-
ters 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=Zkto see how the definitions work). It is convenient to consider
coverings of Dby balls of equal but large radius. Of course, if Dis 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⊆Dbe the set
of centers of the balls in the covering. The idea pursued in this paper is that
if Dis unimodular, then the intensity of Sis a measure of the average number
of points of Sper points of D(Sshould 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 Xis 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
3
an upper bound on the radii. Now, consider a unimodular discrete space D
and a covering of Dby 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 Dis unbounded, then PvR(v)αis always
infinite. The idea is to leverage the unimodularity of Dand to consider the
average of the values R(·)αper point as a replacement of the sum. Under the
unimodularity assumption, this can be defined by E[R(o)α], where ostands 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 rcentered at the
origin and #Nr(o) 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 ras r→ ∞) provide upper and lower bound for the unimodu-
lar 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 Haus-
dorff 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 impor-
tant 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 Haus-
dorff 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
4
discussed in Section 4. These examples are used throughout the paper. Sec-
tion 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 dis-
tribution 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.
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 Ais 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 Dwith distance less than or equal to rfrom v. An
exception is made for r= 0 (Subsection 3.3), where N0(v) := ∅. The diameter
of a subset Ais denoted by diam(A). For a function f: [1,∞)→R≥0, the
polynomial growth rates and polynomial decay rates are defined by the following
5
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 µbe a probability measure on a measurable space Xand
w:X→R≥0be a measurable function. Assume 0 < c := RXw(x)dµ(x)<∞.
By biasing µby wwe 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 Dis 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
Dis a set and oa distinguished point of Dcalled the origin (or the root) of
D. Similarly, a doubly-pointed set is a triple (D, o1, o2), where o1and o2are
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 mis a function m:D×D→Ξ. The mark of a single point
xmay also be defined by m(x) := m(x, x), where the same symbol mis used
for simplicity. An isomorphism (or rooted isomorphism) between two such
spaces (D, o;m) and (D0, o0;m0) is an isometry ρ:D→D0such 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 sim-
ple 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 sim-
ilarly for marked 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.
6
Every element of D∗can be regarded as a boundedly-compact measured met-
ric space (where the measure is the counting measure). Therefore, the gener-
alization 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. Arandom pointed discrete space is a random element in
D∗and is denoted by bold symbols [D,o]. Here, Dand orepresent the discrete
space and the origin respectively.
In this paper, the probability space is not referred to explicitly1. 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 Pand Eare 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 Dand oshould 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 deter-
ministic 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. Arandom pointed marked discrete space is a random
element in D0
∗and is denoted by bold symbols [D,o;m]. Here, D,oand 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.
1Indeed, one may regard D∗, equipped with a probability measure, as the canonical prob-
ability 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∗.
7
2.4 Unimodular Discrete Spaces
Once the notion of random pointed discrete space is defined, the definition
of unimodularity 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 ([·]).
Definition 2.5. Aunimodular 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
Eg+(o)=Eg−(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∈Dchosen 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 uni-
modular 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). There-
fore, unimodular random graphs and networks are special cases of unimodular
(marked) discrete spaces.
Example 2.7 (Point-Stationary Point Processes). Point-stationarity is de-
fined for point processes Φ in Rksuch that 0 ∈Φ a.s. (see e.g., [39]). This
8
definition is equivalent to (2.1), except that gis required to be invariant un-
der translations only (and not under all isometries). This implies that [Φ,0] is
unimodular. In addition, by considering the mark m(x, y) := y−xon 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
Φ can be recovered from [Φ,0; m].
For example, if Φ is a stationary point process in Rk(i.e., its distribution is in-
variant under all translations), with finite intensity (i.e., a finite expected num-
ber 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∈Zis a stochastic process in Rkwith stationary increments such that
X0= 0 and Xi6=Xja.s. for every i6=j, then the image of this random walk
is a point-stationary point process.
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 Dare considered. Intuitively, an
equivariant process on Dis an assignment of (random) marks to Dsuch 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 disinte-
gration 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 Ξ is fixed as in Subsection 2.2.
Definition 2.8. Let Dbe a deterministic discrete space which is boundedly-
finite. A marking of Dis a function from D×Dto Ξ; i.e., an element of ΞD×D.
Arandom marking of Dis a random element of ΞD×D.
Definition 2.9. An equivariant process Zwith values in Ξ is a map that
assigns to every deterministic discrete space Da random marking ZDof D
satisfying the following properties:
(i) Zis 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].
9
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 Dis clear from the context, ZD(·) is also denoted by
Z(·) for simplicity.
Note that in the above definition, Dis 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 PDis the distribution of ZD(for every D) and µis the distribution of
[D,o] (note that only the distribution of ZDis 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 ZDand [D,o;ZD] will be
denoted by the same symbols Pand E; e.g., P[ZD∈B], P[[D, o;ZD]∈A] and
E[f[D,o;ZD]]. The symbols PDand Qwill 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 Zis an equivariant process on Z: Choose U∈ {0,1, . . . , n −1}
uniformly at random and let ZZ(x) := 1 if x∈nZ+Uand 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 Zis 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 ZDdoes 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 Zk
defines an equivariant process on Zk.
Definition 2.14. An equivariant subset Sis 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].
10
For example, SD:= {v∈D: #N1(v) = 4}defines an equivariant subset.
Also, let D=Zand SDbe the set of even numbers with probability pand the
set of odd numbers with probability 1 −p. Then, Sis an equivariant subset of
Zif 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 San 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) stationary 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 frame-
work of unimodular discrete 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 Sub-
section 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 Di-
mensions
This section presents the new notions of dimension for unimodular discrete
spaces. As mentioned in the introduction, the statistical homogeneity of uni-
modular discrete spaces is used to define discrete analogues of the Minkowski
and Hausdorff dimensions. Also, basic properties of these definitions are dis-
cussed.
3.1 The Unimodular Minkowski Dimension
Definition 3.1. Let [D,o] be a unimodular discrete space and r≥0. An
equivariant r-covering Rof Dis an equivariant subset of Dsuch that the
11
set of balls {Nr(v) : v∈RD}cover Dalmost 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 λris non-increasing in terms of r. A smaller λrheuristically 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 dimen-
sions of Dare defined by
udimM(D) := decay (λr),
udimM(D) := decay (λr),
as r→ ∞. If the decay rate of λrexists, 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}kis 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 Dis 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/#Dfrom 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).
12
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.
Sketch of the proof. Let S1,S2, . . . be a sequence of r-coverings of Dsuch 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
Sis an equivariant subset Sof 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 ron 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 Dis K-bounded if
each point of Dis covered at most Ktimes a.s. A sequence (Rn)nof equivariant
coverings of Dis 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)nis a sequence of equivariant coverings which is uniformly bounded,
with Rnan 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 R0be 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] ≤PR0(o)6= 0and 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 Ris 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.
13
Example 3.11. The covering of Zk(equipped with the l∞metric) constructed
in Example 3.3 is a disjoint covering. So it is optimal and hence udimM(Zk) = k.
For Zkequipped with the Euclidean metric, one can construct a 3k-bounded
covering similarly and deduce the same result.
Example 3.12. Let Tkbe the k-regular tree. For r≥1, consider a deterministic
covering of Tkby disjoint balls of radius r. By choosing oin 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 Φon Rendowed
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 ϕbe a discrete subset of R. Let Urbe a random number
in [0, r) chosen uniformly. For each n∈Z, put a ball of radius rcentered
at the largest element of ϕ∩[nr +Ur,(n+ 1)r+Ur). Denote this random
r-covering of ϕby Rϕ. One can see that Ris equivariant under translations
(see Remark 2.13). This implies that Ris an 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 Ris a 3-bounded covering, Lemma 3.9 implies the two left-hand-
side equalities. For all β < decay (p(r)), one has p(r)< r−βfor large enough
r. So, if in addition, β < 1, then q(r)< cr−βfor 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
vrepresents 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 vis defined to be 0. In relation with this convention,
the following notation is used for all discrete spaces Dand points v∈D:
Nr(v) := {u∈D:d(v, u)≤r}, r ≥1,
∅, r = 0.
14
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 Rof Dis 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 Dalmost 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 unimodu-
lar 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,
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 Dis 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 Snof Zkconstructed 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≥0such that ∀r≥1:#Nr(o)≤crαa.s., then udimH(D)≤α.
15
Proof. Let Rbe 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 Ris a covering). By the assumption
and the mass transport principle (2.2), one gets
E[R(o)α]≥1
cE[#NR(o)] = 1
cEg+(o)=1
cEg−(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 principle (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
2and [Z2,0] with proba-
bility 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 uniformly bounded. One has P[R(0) >0] = 1
2(n−1+n−2). This im-
plies 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 Zthan 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 Gnbe the union of a n×nsquare grid (regarded as a graph)
and a path of length n2sharing a vertex with the grid. To cover Gnby balls of
radius r, a fraction of order 1/r of the vertices of Gnare needed (as ris fixed
and n→ ∞). So it is not counterintuitive to say that Gnis one dimensional
16
asymptotically. Indeed, Gntends to the random graph of Example 3.19 in the
local weak convergence [2] as n→ ∞ (if one chooses the root of Gnrandomly
and uniformly).
Remark 3.21. In Example 3.19 above, different samples of Dhave 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]∈Adoes not depend
on the origin of Dand 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≥0 and r≥1,
inf{E[R(o)α] : Ris an equivariant r-covering}=rαλr.
This readily implies that Hα
1(D)≤rαλrfor 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 re-
ferred 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 mono-
tonicity.
17
Definition 3.24. The unimodular α-dimensional Hausdorff size of D(in
short, unimodular 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
Mαis 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).
(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 Ris an equivariant covering, them MRis also an equivariant
covering and satisfies ∀v∈D:MR(v)∈ {0} ∪ [M , ∞) a.s.
(ii). The claim is implied by part (i).
(iii). If Ris 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)=0and 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 βsuch 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 Dcan be zero, finite
or infinite. The lattice Zkprovides 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
Hausdorff 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 Dis finite a.s., this can be proved by putting a
single ball of radius M∨diam(D) centered at a point of Dchosen uniformly
at random. Second, assume Dis infinite a.s. It is enough to construct an
equivariant covering Rsuch that P[R(o)>0] is arbitrarily small. Let p > 0 be
arbitrary and Sbe the Bernoulli equivariant subset obtained by selecting each
point with probability pin an i.i.d. manner. For all infinite discrete spaces D
18
and v∈D, let τD(v) be the closest point of SDto 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 Ris 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 Dis finite with probability q. For all deterministic discrete
spaces D, let RDbe one of the above coverings depending on whether Dis finite
or infinite. It satisfies P[R(o)>0] = E[1/#D] + p(1 −q). Since pis 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 Snbe the covering in Example 3.3 scaled by factor δ. One has
ESn(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
cis 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.
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 dis the metric on Dand ois the origin. Note
that if d0is another metric on D, then d0∈RD×D. So d0can be considered
as a marking of Din 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
equivariant process d0such 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 bound-
edly finite metric space.
If in addition, [(D,d),o] is a unimodular discrete space, then [(D,d),o;d0]
is a unimodular 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 d0be an equivariant metric. If d0≤cd+aa.s., with cand aconstants,
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).
19
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+aa.s., then (D,d0) has the same
unimodular dimensions as (D,d). Also, cDhas the same dimension as Dand
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 di-
mensions 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 Gare 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 lis an
equivariant process which assigns a positive weight to the edges of every deter-
ministic graph, then one can let d0(u, v) be the minimum weight of the paths
that connect uto v. If d0is 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 Sbe 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 Sas above, the unimodular Haus-
dorff 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 San equivariant
subset such that SDis nonempty a.s. Then,
(i) One has
udimH(SD) = udimH(D),
udimM(SD)≥udimM(D),
udimM(SD)≥udimM(D).
(ii) If ρis the intensity of Sin D, then for every α≥0, the α-dim H-size of
SDsatisfies
2−αρMα(D)≤ Mα(SD)≤ρMα(D).
20
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
Dis regular (see Remark 3.23), which immediately follows from Theorems 3.22
and 3.34. Also, equality is guaranteed if SDis a r-covering of Dfor 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 Rbe 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 R0is a 2r-covering of SD. One has
Po∈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 Rnof equivariant coverings of SDsuch 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 Dby letting Rn(v) := 0 for v∈D\SD. Let > 0 be
arbitrary and Bn⊆Dbe the union of N(1+)Rn(v) for all v∈D. Define
R0
n(u) := (1 + )Rn(u) for u∈Bnand R0
n(u) := 1/ for u6∈ Bn. It is clear
that R0
nis an equivariant covering of D. Also,
ER0
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)
21
Since the radii of the balls in Rnare at least n, one gets that Bnincludes the
n-neighborhood of SD. Therefore, P[o6∈ Bn]≤P[Nn(o)∩SD=∅]. Since
SDis nonempty a.s., this in turn implies that P[o6∈ Bn]→0 as n→ ∞ (note
that the events Nn(o)∩SD=∅are nested and converge to the event SD=∅).
So (3.7) implies that
lim inf
n→∞
ER0
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
nare 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 Rnbe a sequence of equivariant coverings of Dfor 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
nis an equivariant covering of SD. One has
ER0
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→∞
ER0
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 considering 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 Cis
defined (which is an equivariant collection of finite subsets), one can define the
average diameter of sets U∈Cper 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#),
22
where the infimum is over all equivariant coverings C. Here, taking the maxi-
mum with Mis similar to the condition that the subsets have diameter at least
2M(note that a ball of radius Mmight 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).
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 λrby 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 equiv-
ariant 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 un-
ordered). To show the idea, consider a covering C={U1, U2, . . .}of a determin-
istic discrete space D, where each Uiis 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 uni-
formly. 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 Ξ to 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 con-
tinuum 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 of-
ten, 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 Haus-
dorff dimension by using dyadic cubes instead of balls. This changes the value
23
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 uni-
modular 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 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. In addition, It will be shown in Subsection 6.2 that unimodular trees
with infinitely many ends have exponential volume growth, and hence, have
infinite Hausdorff dimension. The remaining two cases are discussed below.
4.1.1 Unimodular Two-Ended Trees
If Tis a tree with two ends, then there is a unique bi-infinite path in Tcalled 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
ρis 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 STbe the trunk of T. Then, Sis an equiv-
ariant subset. Therefore, Theorem 3.34 implies that udimH(T) = udimH(ST).
Since the trunk is isometric to Zas 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.
24
The claim concerning the unimodular Minkowski dimension is implied by
Corollary 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 par-
ticular, 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 vis 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 should 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. It is not known whether the first inequality in (4.3) is always
an equality.
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].
The cone with height nat v∈Tis defined by Cn(v) := Nn(v)∩D(v); i.e.,
the first ngenerations of the descendants of v, including vitself. Let λ0
nbe 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 nis also a covering by
balls of radius n. This implies that λn≤λ0
n. Also, if Sis a covering by balls
25
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 Sof 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 vor not. So the
output of the algorithm is well defined.
Proof. Let Abe any equivariant covering of Tby cones of height n. Consider
a realization (T;A) of [T;A]. Let vbe a vertex such that h(v) = n. Since Ais
a covering by cones of height n,Ashould have at least one vertex in D(v) (to
see this, consider the farthest leaf from vin D(v)). Now, for all such vertices
v, delete the vertices in A∩D(v) from Aand then add vto A. Let A1be
the subset of Tobtained by doing this operation for all vertices vof height n.
So A1is also a covering of Tby cones of height n. Now, remove all vertices
{v:h(v) = n}and their descendants from Tto 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 Tsuch that,
for each i,Aiis a covering of Tby cones of height nwhich agrees with STon
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 equiv-
ariant subsets A=A0,A1, . . . on T. It can be seen that the intensity of
A1is at most that of A(this can be verified by the mass transport princi-
ple (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=Sas equivariant subsets of T.
This implies that P[o∈A]≥P[o∈S], hence, Sis 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
nis a covering of Tby cones
of height 2n−2. Let v∈Tbe an arbitrary vertex. Let kbe the unique integer
such that (k−1)n−1< h(v)≤kn −1. Let jbe the first nonnegative integer
26
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 wto 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 vis covered by the cone of height 2n−2 at Fn−1(z).
Since Fn−1(z)∈B0
n, it is proved that B0
ngives 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 Sbe the optimal covering by cones of
height ngiven by the algorithm of Lemma 4.3. Send unit mass from each vertex
v∈Sto the first vertex in v, F (v), . . . , F n(v) which belongs to Bn+1 (if there
is any). So the outgoing mass from vis 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∈Sand the claim is proved. So assume
h(w)> n. By considering the longest path from win D(w), one can find a
vertex zsuch 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 witself lies in S. Note that in the
algorithm in Lemma 4.3, at each step, the height of wdecreases by a value at
least 1 and at most n+ 1 until wis removed from the tree. So in the last step
before wis removed, the height of wis 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 Tare 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 vto F(v) if h(v) = nand 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).
27
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 n0< n2<··· such that P[h(o) = ni]< n−α
ifor 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.
.
By arguments similar to Lemma 4.4, it can be seen that Rkis indeed a 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
ERk(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−αover
x≥ni−1happens at α
ni−1and 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 Ckwith offspring cardinality k[1] is constructed as follows. Its
vertex set is partitioned in levels L0, L1, . . .. Each vertex in level nis connected
to kvertices in level n−1 (if n6= 0) and one vertex (its parent) in level n+ 1.
Let obe a random vertex of Cksuch 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)
28
is the height of vdefined in Subsection 4.1.2. The set Sgives 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 Lnand Ln+1 be an,
where a > 1 is constant. Let d1be the resulting metric on Ck. Given r > 0, let
S1be the set of vertices having distance at least r/a to L0(under d1). One can
show that S1is 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 ancentered 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 anby n! in the second case and let d2be the resulting metric.
Then, the cardinality of the ball of radius rcentered at ohas 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 pro-
vide an example where the lower Minkowski dimension, the upper Minkowski
dimension and the Hausdorff dimension are all different when suitable parame-
ters are chosen.
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)∈Φnand Φ := ∪iΦi. Then, Φ is a point
process in the plane which is stationary under horizontal translations. Choose
mindependent of the sequence (Ui)isuch that P[m=n] = pnfor each n.
Then, let o:= om.
Construct a graph Ton Φ as follows: For each n, connect each x∈Φnto
its closest point (or closest point on its right) in Φn+1. Note that Tis 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 unimodular tree [T,o] constructed above.
Note that in the case where pnis proportional to k−nfor kfixed, [T,o] is
just the ordinary canopy tree Ckof Subsection 4.2.1. Also, one can generalize
the above construction by letting Φnbe 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.
29
Proof. For part (i), it is enough to show that Φ −ois 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−onis 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 Tcan be realized as an equivariant process on Φ (see
Definition 2.9 and Remark 2.13). Therefore, by Lemma 2.12 and Theorem 3.31,
it is enough to prove that Tis connected a.s. Nevertheless, the same lemma
implies that the connected component T0of Tcontaining ois a unimodular
tree. Since it is one-ended, Theorem 3.9 of [5] implies that the foils T0∩Φiare
infinite a.s. By noting that the edges do not cross (as segments in the plane),
one obtains that T0∩Φishould be the whole Φi; hence, T0=T. Therefore, T
is 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 Tis 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. Tis 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 Φnfor all n > 0. Therefore, for all n≥0, the height of every vertex in Φnis
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).
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−αand has slope larger
than −n−γ. Now, let qni:= n−β
ifor each iand define qnlinearly in the interval
[ni, ni+1]. Let pn:= qn−qn+1. It can be seen that pnis non-increasing,
decay (qn)≤α, decay (qn) = βand decay (pn)≥γ.
30
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 proba-
bility 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 equal-
ity 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 µbe a probability measure on Rk. Consider the (double-sided) simple
random walk (Sn)n∈Zin Rkstarting from S0:= 0 such that and the jumps
Sn−Sn−1are 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∈Zis a random discrete subset
of Rk. If no point of Rkis visited more than once (e.g., when Snis 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, Φ is actually a point stationary renewal process [24].
Proposition 4.9. Let Φ := {Sn}n∈Zbe the image of a simple random walk S
in Rstarting from S0:= 0. Assume the jumps Sn−Sn−1are positive a.s. Then
31
(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]).
(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 Propo-
sition 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 Zkwill 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 Tis transient a.s., then the image of the (two sided) simple
random walk on Tis 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 Ψbe the zero set of the symmetric simple random walk
on Zwith 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)nis another simple random walk and Ψ is its
image. The distribution of the jump S1is 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
2for 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 ⊆Rbe an arbitrary point-stationary point process. Let S1be 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)αeequal parts. Let Φ0denote 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 Φ0by dSα
1eand 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
32
in [2]), (it is not a renewal process). The distribution of Ψ is determined by the
following equation, where his 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 Φand Ψbe as above. Then, Φhas 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 proposition implies udimM(Ψ) <udimH(Ψ).
Proof. Let Abe 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 \Aconditioned on 0 6∈ Ahas 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
cEdSα
1e1{Φ0∩(0,r)=∅}
=1
cEhdSα
1e1{S1/dSα
1e>r}i.
Now, by the assumption decay (P[S1> r]) = βand integration by parts, it
is straightforward to deduce that decay (P[Ψ ∩(0, r) = 0]) = (β−α)/(1 −α).
Therefore, Proposition 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
33
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 Tis
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 Tis 2. In the following, the graph-distance metric is considered on T.
Proposition 4.13. One has udimM(T) = udimH(T) = 3
2under 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. Follow-
ing [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 re-
sults on random walks imply that P[h(o)≥n] is bounded between two constant
multiples of n−1
2for all n. So Theorem 4.2 implies that udimM(T) = 3
2.
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, . . . , flbe similitudes of Rk
with similarity ratios r1, . . . , rlrespectively (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 Rkinstead 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, Knconverges 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⊆Rksuch that K=Sifi(K).
In addition, if the fi’s satisfy the open set condition; i.e., there is a bounded
open set V⊆Rksuch that fi(V)⊆Vand fi(V)∩fj(V) = ∅for each i, j,
then the Minkowski and Hausdorff dimensions of Kare equal to the similarity
dimension, which is the unique α≥0 such 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.
34
Theorem 4.14. Let onbe a point of Knchosen uniformly at random, where
Knis defined in (4.9). Assume that ri=r < 1for all iand 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 Rkwill be constructed such that [r−nKn,on] converges weakly to [Ψ, o]. In
addition, Ψ −ois 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 Zkand 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, Knis 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)⊆Zconverges weakly to
the random set Ψ ⊆Zdefined as follows: Ψ := ∪nTn, where Tnis 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 Tnhas the same
distribution as Ψn, but the sequence Tnis nested. In addition, since onis chosen
uniformly, Ψnand 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 Cnbe the set of
points in the n-th step of the construction of the Koch snowflake. Let xnbe a
random point of Cnchosen uniformly and Φn:= 3n(Cn−xn). It can be seen
that Φntends 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.
35
Figure 1: Four ways to attach 3 isometric copies to Tnin 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.
In addition, Φ can be constructed explicitly as Φ := ∪nTn, where Tnis a
random finite path in the triangular lattice with distinguished end points Anand
Bndefined inductively as follows: Let T1:= {A1,B1}, where A1is the origin
and B1is 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 Tnthree 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 Tnrelative 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 anbe the distance of onto its closest point in Kn,
it is shown that for any > 0, P[an/(¯r)n< ]→1
2and Pan/(¯r)n>1
→1
2,
where ¯ris 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 Knis not necessarily contained in Kn+1, unless ois 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 δnensures that ˆ
Kn⊆ˆ
Kn+1 for all n. It is
36
easy to see that [ ˆ
Kn, o] has the same 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|=nfor each j≤l. Let
Dbe a fixed number such that Vintersects BD(o). Now, the sets fσj(V) for
1≤j≤lare disjoint and intersect a common ball of radius Drn. Moreover,
each of them contains a ball of radius arnand 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(·)≤Ca.s., hence w(·)≤Ca.s.
(ii). Let Dbe 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)ka.s. Since this holds for all large enough D, one
obtains that ˆ
Kis a discrete set a.s.
(iii). Note that the distribution of o0
nis just the distribution of onbiased
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 onis 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
Kmeans the dimension
of the unimodular space obtained by biasing the distribution of ˆ
Kby 1/w(o)
(see Lemma 4.20). Let D > diam(K) be given, where Kis the attractor of
f1, . . . , fl. Let m > 0 be large enough so that diam(Km)< D. Note that
each element in ˆ
Kcan be written as f−1
δnfσ(o) for some nand some string σ
of length n. Let γmbe a string of length mchosen 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).
37
Note that Uσ⊆ˆ
Kis always a scaling of Kmwith ratio r−mand 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 Rmgives an equivariant covering. Also, note that Rm(o)>0
if and only if fσfγm(o) = fδn+m(o) for some nand some string σof length n. Let
An,m(o) be the set of possible outcomes for γmsuch that there exists a string
σof length nsuch that the last equation holds. One can see that this set is
increasing with nand 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
E1
w(o)1{Rm(o)>0}=Ew0
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
E1
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 msuch that r−m≤D < r−m−1. By Lemma 4.20,
there are finitely many points in ˆ
K∩BD(o). Therefore, one finds nsuch 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, . . . , σkare strings of length nsuch 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 Vcontains 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
38
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 unimodular self similar discrete space can be constructed by successively
attaching copies of a set to itself. This is expressed in the following algorithm.
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
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 ex-
amples. 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 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.
39
5 The Unimodular Mass Distribution Principle
and Billingsley Lemma
Let Dbe 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.
Dhas 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≥0will 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 wis an equivariant process
(Definition 2.5) with values in R≥0. For all discrete spaces Dand 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 uni-
modular.
5.1 Unimodular Mass Distribution Principle
Theorem 5.2 (Mass Distribution Principle).Let [D,o]be a unimodular dis-
crete space.
(i) Let α, c, M > 0and 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)] .
40
(ii) If in addition, wD(o)>0with positive probability, then udimH(D)≤α.
Proof. Let Rbe 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 wand R; i.e., for each deterministic dis-
crete space G, choose wGand RGindependently (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 Ris a covering. Therefore, the mass trans-
port principle implies that E[w(NR(o))] ≥E[w(o)] (recall that by convention,
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)>0 and 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≥0and wis 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]≥ER(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 vbe an arbitrary point in NR(o)∩Am. By
the definition of Am, one gets that for all r≥m,w(Nr(v)) ≤c0rα.Since
41
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
2αc0Ew(NR(o))1{NR(o)∩Am6=∅}.(5.4)
By letting g(u, v) := w(v)1{v∈NR(u)}1{NR(u)∩Am6=∅}, one gets that g+(o) =
w(NR(o))1{NR(o)∩Am6=∅}. Also, since there is a ball NR(u) that covers oa.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
2αc0Ew(o)1{o∈Am}.This implies that
Hα
m(D)≥1
2αc0Ew(o)1{o∈Am}. Using (5.2) and letting mtend to infinity
gives 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
Theorem 5.2. The proof leverages Lemma 3.26.
Lemma 5.4 (Lower Bounds).Let [D,o]be a unimodular discrete space, α≥
0and 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 infr→∞ P[w(Nr(o)) ≤δrα]=0, then udimH(D)≥α.
(iv) If decay Ehexp −w(Nn(o))
nαi≥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,γand κbe such that γ < β < κ < α. Fix n∈N. Let S=SDbe
the equivariant subset obtained by selecting each point v∈Dwith probability
1∧(n−βw(v)) (the selection variables are assumed to be conditionally indepen-
dent given [D,o;w]). Let Rn(v) = nif v∈SD,Rn(v) = 1 if Nn(v)∩SD=∅,
and Rn(v) = 0 otherwise. Then Rnis 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=∅]
=nγE1∧n−βw(o)+E
Y
v∈Nn(o)1−(1 ∧n−βw(v))
≤nγ−βE[w(o)] + Eexp −n−βw(Nn(o))
=nγ−βE[w(o)] + Eexp −n−βw(Nn(o))|M < n P[M < n]
+Eexp −n−βw(Nn(o))|M≥nP[M≥n]
≤nγ−βE[w(o)] + exp −nκ−β+P[M≥n],
42
where the first inequality holds because 1−(1∧x)≤e−xfor all x≥0. Therefore,
E[Rn(o)γ]→0 when n→ ∞. It follows that udimH(D)≥γ. Since γis
arbitrary, this implies udimH(D)≥α.
(i). Only a small change is needed in the above proof. For n≥r0, let
Rn(v) = nif either v∈SDor Nn(v)∩SD=∅, and let Rn(v) = 0 otherwise.
Note that Rnis 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−βfor large n. This implies that udimM(D)≥β. Since βis
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 Eexp −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 Rnsuch that ERn(o)β<∞and (iii) is proved.
Let > 0 be arbitrary. By the assumption, there is δ > 0 and r≥1 such
that P[w(Nr(o)) ≤δrα]< . So
Eexp −r−βw(Nr(o)) ≤Eexp −r−βw(Nr(o))|w(Nr(o)) > δrα
+P[w(Nr(o)) ≤δrα]
≤exp(−δrα−β) + .
Note that for fixed and δas above, rcan be arbitrarily large. Now, choose
rlarge enough for the right hand side to be at most 2. This shows that
Eexp −r−βw(Nr(o))can be arbitrarily small and the claim is proved.
(iv). As before, let Rn(v) = nif either v∈SDor 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)] + Eexp −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→∞
Eexp −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 wsuch 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))]) .
43
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)) ≤rαfor 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 because 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 wequal 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.
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 Exam-
ple 2.7). For v∈Φ, let w(v) be the sum of the distances of vto 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 Subsection 4.3.1.
Also, the condition that E[wD(o)] >0 is trivially necessary for the upper
bound.
Corollary 5.10. Let [G,o]be a unimodular random graph equipped with the
graph-distance metric. If Gis infinite almost surely, then udimM(G)≥1and
else, udimM(G) = udimH(G)=0.
Proof. If Gis infinite a.s., then for wG≡1, one has w(Nr(o)) ≥rfor 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
unimodular two-ended tree are equal to one.
44
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 quan-
tities 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, unimod-
ularity is, roughly speaking, equivalent to point-stationarity. To study the di-
mension of such processes, the following covering is used in the next results.
Let ϕbe a discrete subset of Rkequipped with the l∞metric and r≥1. Let
C:= Cr:= [0, r)k,U:= Urbe a point chosen uniformly at random in −C,
and consider the partition {C+U+z:z∈rZk}of Rkby cubes. Then, for
each z∈rZk, choose a random element in (C+U+z)∩ϕindependently (if
the intersection is nonempty). The distribution of this random element should
depend on the set (C+U+z)∩ϕin a translation-invariant way (e.g., choose
with the uniform distribution or choose the least point in the lexicographic or-
der). Let R=Rϕassign the value rto the selected points and zero to the other
points of ϕ. Then, Ris an equivariant covering. Also, each point is covered at
most 3ktimes. So Ris 3k-bounded (Definition 3.8).
Theorem 5.13 (Minkowski Dimension in the Euclidean Case).Let Φbe a
point-stationary point process in Rkand assume the metric on Φis 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))]) ,
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 vwith probability wϕ(v)/wϕ(Cr+Ur+z) (conditioned on wϕ). One gets
P[0 ∈R] = E[w(0)/w(Cr+Ur)] .As mentioned above, Ris equivariant and
uniformly bounded (for all r > 0). So Lemma 3.9 implies both equalities in the
45
claim. The inequalities are implied by the facts that w(Cr+Ur)≤w(Nr(0))
and
Ew(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∈Zbe 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 Φ0be the union of these points together with Z. Bias the distribution
of Φ0by T0(Definition 2.1) and then translate Φ0by 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 ET2
0=∞, one gets
E[Nr(0)] ≥E[N1(0)] = ∞. But one can show that decay (E[1/Nr(0)]) = 1.
Proposition 5.15. If Φis a point-stationary point process in Rkand the metric
on Φis equivalent to the Euclidean metric, then udimH(Φ) ≤k.
Proof. One may assume the metric on Φ is the l∞metric without loss of gener-
ality. Let C:= [0,1)kand Ube a random point in −Cchosen uniformly. For all
discrete subsets ϕ⊆Rkand v∈ϕ, let C(v) be the cube containing vof the form
C+U+z(for z∈Zk) and wϕ(v) := 1/#(ϕ∩C(v)). Now, wis 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 Billings-
ley lemma (Theorem 5.6) implies that udimH(Φ) ≤k.
Proposition 5.16. If Ψis a stationary point process in Rkwith finite intensity
and Ψ0is 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 Ψ0be a shifted square lattice independent
of Ψ (i.e., Ψ0=Zk+U, where U∈[0,1)kis 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
0of Ψ00 is obtained by the superposition
of Ψ0and an independent stationary lattice with probability p(heads), and the
superposition of Zkand Ψ with probability 1−p(tails). So part (i) of Lemma 5.4
implies that udimM(Ψ00
0)≥k. Note that Ψ00
0has two natural equivariant subsets
which, after conditioning to contain the origin, have the same distributions
as Ψ0and Zkrespectively. Therefore, one can use Theorem 3.34 to deduce
46
that udimM(Ψ0)≥udimM(Ψ00
0) = k. Therefore, Proposition 5.15 implies that
udimH(Ψ0) = udimM(Ψ0) = k.
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 Proposi-
tion 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 Φis a point-stationary point process in Rkwhich 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 −Snis 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 equiv-
ariant weight functions w1and w2such 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 > 0 be arbitrary and
A:= {[D, o]∈ D∗: growth (w1(Nr(o))) >growth (w2(Nr(o))) + }.
It can be seen that Ais 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 Adoes not depend on the root (i.e., if [D, o]∈A, then ∀v∈
47
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]∈Aa.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 Dis either Z,
the Palm version of a stationary point process in Rkor a point-stationary point
process in R, Birkhoff’s pointwise ergodic theorem (or its generalizations) im-
plies 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 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 func-
tions seems more interesting.
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 was
long as 0<E[w(o)] <∞?
5.5 Notes and Bibliographical Comments
As already mentioned, the unimodular mass distribution principle and the uni-
modular Billingsley lemma have analogues in the continuum setting (see e.g.,
[14]) and are named accordingly. Note however that there is no direct or sys-
tematic reduction to these continuum results. For instance, in the continuum
setting, one should assume that the space under study is a subset of the Eu-
clidean 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 con-
trast 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.
48
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 Subsection 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)] ≤cn2for a constant
c. Recall from Subsection 4.1.2 that F(v) represents the parent of vertex vand
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], Ynis a critical
Galton-Watson tree up to generation n. Also, for 0 ≤i < n,Yihas the same
structure up to generation i, except that the distribution of the first generation
is size-biased minus one (i.e., (npn+1)nwith 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 Sub-
section 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−βfor
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 ra.s. (for the proof, see
[7] or Theorem 4 of [26]). Therefore, the unimodular Billingsley lemma (Theo-
rem 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 rfor large enough r. Then, part (iii) of the proposition implies that
udimH(Φ) = 0. However, since Φ is 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
2is proved
in Subsection 4.5. So it remains to prove udimH(T)≤3
2. To use the uni-
modular Billingsley 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 kof Fl(o) which are not a descendant of Fl−1(o) (for l= 0, let it be just
49
#F−k(o)). One has #Nn(o) = Pk,l ek,l1{k+l≤n}. It can be seen that E[ek,l ] is
equal to the probability that two independent paths of length kand lstarting
both at odo not collide at another point. Therefore, E[ek,l ]≤c(k∧l)−1
2for
some cand all k, l. This implies that (in the following, cis 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
2for some cand all n. There-
fore, 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 equivari-
ant 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 Tcan 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 pcof percolation on Tis 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, pcis 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) = ∞.
The following example shows that the Minkowski dimension can be finite.
50
Example 6.5. Let Tbe the 3-regular tree. Split each edge eby adding a
random number leof new vertices and let T0be the resulting tree. Let vebe
the middle vertex in this edge (assuming leis always odd) and assign marks
by m0(ve) := le. Assume that the random variables leare 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 Rbe 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 trans-
port principle, one can show that ρ(R)≥ρ(Ar), where ρ(·) = P[o∈ ·] denotes
the intensity. On the other hand, let Sbe the equivariant subset of vertices
with degree 3. Send unit mass from every point of Arto 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 Sbe the equivariant subset of vertices of degree at least 3.
For every v∈S, let w(v)be the sum of the distances of vto 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
Subsection 6.3.3 as well.
Proposition 6.7. Let [T,o]be a unimodular tree such that the degree of ev-
ery vertex is at least 3. Let d0be an equivariant metric on T. Let w(v) :=
Pud0(v, u), where the sum is over the 3 neighbors of vwhich 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 d0is 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∈Tand 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 vin S. For v∈T\S, if u1and u2are the two closest
points of Sto v, let g(v, ui) := d(ui, v)α−1and 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)) ≥crαfor all v∈Tand r≥0 (see Lemma C.5 in [7]) and the claim
is proved.
51
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.
Choose the sequence (pn)nin the definition of [T,o] such that pn=c2−qn
and Pnpn= 1, where cis constant and q0≤q1≤ · ·· is a sequence of integers.
In this case, Tis 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 vis a vertex in the n-th level of T, then the number of
descendants of vis (p0+·· · +pn)/pn. It follows that growth(T) = decay (pn)
and growth (T) = decay (pn). So, by choosing (pn)nappropriately, Tcan 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 dimen-
sional (note that this tree differs from the Eternal Galton-Watson tree of Sub-
section 4.2.3 which is a directed tree). Consider an ordinary Galton-Watson tree
with offspring distribution µ= (p0, p1, . . .), where µis 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 Tis
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 Tis 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
conditioned on non-extinction. Then, udimM(T) = udimH(T) = ∞.
Proof. The result for the Hausdorff dimension follows from the unimodular
Billingsley lemma (Theorem 5.6) and the Kesten-Stigum theorem [34], which im-
plies that limn#Nn(o)m−nexists 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].
52
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) := Pnpnsnbe the generating function of µ. By classical results
of the theory of branching processes, for all s≤1, Esdn(e
o)=f(n)(s),where
dn(˜
o) := #Nn(e
o)−#Nn−1(e
o) and f(n)is the n-fold composition of fwith 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 fis convex. Therefore,
acan 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.
It follows that decay f(n)(1 −n−α)=∞. So 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
Tas 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≥0with 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 oby 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∈Zin Rk, where
S0= 0 and the increments Sn−Sn−1are i.i.d. The graph of the random walk
53
(Sn)n∈Zis 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 S1has 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 pre-
served and dimension is not decreased. Under this metric, the ball Nn(0) is
Ψ∩[−n2, n2]×[−n, n]. It is straightforward that Z2has volume growth rate 3
and also Minkowski and Hausdorff dimension 3 under this metric.
Proposition 6.10. If the jumps are ±1uniformly, under the metric (6.1), the
graph Ψof 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 Cbe the equivariant disjoint covering of Z2by 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=SΨbe the set of
selected points. By construction, Sgives an n-covering of Ψ and it can be
seen that it is an equivariant covering. Let σ0be the rectangle containing the
origin. By construction, 0 ∈Sif and only if it is either on a right-edge of σ0
or on a horizontal edge of σ0and 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 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
54
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 Jin Z≥0are defined by d(J) :=
lim supn1
nJnand d(J) := lim inf n1
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 Tnby a ball of radius 2ncentered 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 Rnis 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 Enis defined by
the following random algorithm: Let E0:= [0,1]k. For each n≥0 and each
b-adic cube of edge length b−nin En, divide it into bksmaller b-adic cubes of
edge length b−n−1. Keep each smaller b-adic cube with probability pand 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−k
and otherwise, has dimension k+ logbpconditioned on being non-empty.
For each n≥0, let Knbe 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 Haus-
dorff metric.
Proposition 6.12. Let K0
ndenote the random set obtained by biasing the dis-
tribution 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].
55
(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
Kwill be used.
First, consider the usual nested sequence of partitions Πnof Zkby 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}kbe i.i.d. uniform numbers
and let Un=Pn
i=0 aibi∈Zk. Shift the partition Πnby the vector Unto 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)nbe the stationary nested sequence of partitions of Zk
defined above. For each n≥0and each cube C∈Π0
nthat does not contain the
origin, with probability 1−p(independently for different choices of C), mark all
points in C∩Zkfor 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 Cnbe the cube in Π0
nthat 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]kbe 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] = PCn=bn([0,1)k−v),
which is left to the reader.
Here is another description of ˆ
K. The nested structure of SnΠ0
ndefines a
tree as follows. The set of vertices is SnΠ0
n. For each n≥0, connect (the vertex
corresponding to) every cube in Π0
nto 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 pand remove it with probability 1−pin an i.i.d. manner.
Let Tbe the connected component of the remaining graph that contains the
root. Conditioned on the event that Tis infinite, ˆ
Kcorresponds 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 Cbe any cube in
Π0
ithat does not contain the origin. Note that the subtree of descendants of C
56
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 lmnfor some constant lnot depending on n.
Note that Nbn(o) is contained in the union of Cnand 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 (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 consid-
ering the partition Π0
nby cubes, one can construct a bn-covering Rnas 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 T0as 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 prob-
ability of the descendants of the root, which is positive. By noticing the fact
that the radii of the balls are bnand the covering is uniformly bounded, one
gets that udimM(ˆ
K) = logbm=k+ logbp.
Finally, it remains to prove that ˆ
Kis 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 Hwith 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 ois 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,
57
(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)> rβfor 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 com-
ments 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 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≥0which 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 cis 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 Rcan be regarded a special case of this construction by letting
cr(v) be 1 when r=R(v) and 0 otherwise.
2The term ‘weighted’ refers to the weighted sums in Definition 7.1 and should not be
confused with equivariant weight functions of Definition 5.1
58
Definition 7.1. Let f:D∗→Rbe 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 mea-
surable 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≥0such
that E[w(o)] = ξα
M(1) and almost surely,
∀v∈D,∀r≥M, w(Nr(v)) ≤rα.(7.3)
(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 wis an equivariant weight function satis-
fying (7.3), then E[w(o)] ≤ξα
M(1) and E[w(o)h(o)] ≤ξα
M(h). Therefore, the
(deterministic) weight function wgiven 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).
59
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 distribution principle (Theorem 5.2) is enough for bounding the Hausdorff
dimension from above. However, there are very few examples in which the
function wgiven by the unimodular Frostman lemma can be explicitly computed
(in some of the examples, a function wsatisfying 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)→Ris continuous. In fact, it is
Mα-Lipschitz; i.e., |ξα
M(f1)−ξα
M(f2)| ≤ MαE[|f1(o)−f2(o)|].
Proof. Let cbe an equivariant weighted collection of balls satisfying (7.1) for f1.
Intuitively, add a ball of radius Mat each point vwith cost |f2(v)−f1(v)|. More
precisely, let c0
r(v) := cr(v) for r6=Mand 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 cis 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 > 0a.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 fand 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)→Rsuch that l(h) = ξα
M(h) and −ξα
M(−f)≤l(f)≤ξα
M(f),for
all f∈L1. Since lis sandwiched between two functions which are continuous
at 0 and are equal at 0 (since ξα
M(0) = 0), one gets that lis continuous at 0.
Since lis linear, this implies that lis continuous. Since the dual of L1(D∗, µ)
is L∞(D∗, µ), one obtains that there is a function w∈L∞(D∗, µ) such that
60
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 wwhich is nonnegative everywhere.
The claim is that wsatisfies the requirements.
Let r≥Mbe fixed. For all discrete spaces D, let S:= SD:= {v∈D:
w(Nr(v)) > rα}. By the definition of SD, one has
Ew(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
Ew(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 collec-
tion of balls for fr: put balls of radius rwith 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≥Msimultaneously. By monotonicity of w(Nr(v))
w.r.t. r, one gets that the latter almost surely holds for all r≥Mas de-
sired. Also, one has E[w(o)h(o)] = l(h) = ξα
M(h).Thus, wsatisfies 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
wis 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) = 0 if and only if Hα
M(D) = 0.
Proof. (i). The first inequality is easily obtained from the definition of ξα
1(1)
by considering the cases where c(·)∈ {0,1}. In particular, this implies that
61
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 cthat satisfies (7.1) for hand EhPr≥Mcr(o)rαi≤b0.Next,
given a≥0, define an equivariant covering Ras follows. For each v∈Dand
r≥Msuch that cr(v)>0, put a ball of radius rat vwith probability acr(v)∧1.
Do this independently for all vand r(one should condition on Dfirst). If more
than one ball is put at v, keep only the one with maximum radius. Let Sbe
the union of the chosen balls. For u∈D\S, put a ball of radius Mat 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 ois 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]≤Ee−ah(o). Thus, (7.6) implies that
E[R(o)]α≤MαEe−ah(o)+ab0. Since a≥0 and 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 Ξ0be 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 Ξ0is 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.
62
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 eis well defined and call it the conductance
of e. Let Lbe the set of leaves of T. As in Subsection 4.1.2, let F(v) be the
parent of vertex vand D(v) be the descendants subtree of v.
Definition 7.10. Alegal equivariant flow on [T;c] is an equivariant way
of assigning extra marks f(·)∈Rto the edges (see Definition 2.9 and Re-
mark 2.13), such that almost surely,
(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 vto 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 Tcan be uniquely represented as (v, F (v)), the
following convention is helpful.
63
Convention 7.11. For the vertices vof 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|:= Ef(o)1{o∈L}.
Also, for the equivariant cut-set Π, define
c(Π) := Ec(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 Π is called equivariantly minimal if there is no
other equivariant 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 fis a legal equivariant flow and Πis an equivariant cut-set,
then |f| ≤ c(Π). Moreover, if the pair (f,Π) is equivariant, then
|f| ≤ Ef(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 vsuch that (v, F (v)) ∈Π. Then, send mass f(v) from each leaf vto τ(v).
By the mass transport principle (2.2), one gets
Ef(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)
=Ef(o)1{o∈Π},
where the last equality holds because fis 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 unimodular marked one-ended tree [T,o;c]equipped with conductances c
as above, if cis bounded on the set of leaves, then
max
f|f|= inf
Πc(Π),
64
where the maximum is over all legal equivariant flows fand the infimum is over
all equivariant cut-sets Π.
Remark 7.15. The claim of Theorem 7.14 is still valid if the probability mea-
sure (the distribution of [T,o;c]) is replaced by any (possibly infinite) measure
Pon 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 Tnbe the sub-forest of Tobtained by
keeping only vertices of height at most nin T. Each connected component of
Tnis a finite tree which contains some leaves of T. For each such component,
namely T0, do the following: if T0has more than one vertex, consider the
maximum flow on T0between 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 T0has 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 fnon the edges and
a cut-set Π0
nare obtained (by letting fnbe zero on the other edges). Π0
nis
always a cut-set, but fnis not a flow. However, fnsatisfies (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 T0of 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 uis 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
Efn(o)1{o∈L}=E[h(o)] =Ec(o)1{o∈Π0
n}=c(Π0
n).
Since 0 ≤fn(·)≤cn(·), one can see that the distributions of fnare 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 f0is a legal equiv-
ariant flow. Also, since f0(o) 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=Ef0(o)1{o∈L}= lim
i
Efni(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.
65
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 Φbe a point-stationary point process in Rkendowed with
the l∞metric, and let α≥0. Then, there exists an equivariant weight function
won Φsuch 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). For every integer n≥0,
let Qnbe the stationary partition of Rkby translations of the cube [0, bn)kas
in Subsection 5.3. Consider the nested coupling of these partitions for n≥0
(i.e., every cube of Qnis contained in some cube of Qn+1 for every n≥0)
independent of Φ. Let T0be the tree whose vertices are the cubes in ∪nQn
and the edges are between all pairs of nested cubes in Qnand Qn+1 for all n.
Let T⊆T0be the subtree consisting of the cubes qn(v) for all v∈Φ and
n≥0. The set Lof the leaves of Tconsists 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∈Lcorresponds 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, gcan be allowed to depend on Tin
this equation (but the sum is still on σ0∈L). Therefore, one can assume the
metric on Lis 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 Tunimodular. 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)
66
where en:= Φ(qn(0)). Let Edenote the integral operator w.r.t. the measure
P. For any equivariant flow fon T, the norm of fw.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}
=E1
Φ(σ)f(σ),
where the second equality is by (7.10). Consider the conductance function
c(τ) := bnα for all cubes τof edge length bnin Tand all n. Therefore, Theo-
rem 7.14 and Remark 7.15 imply that the maximum of E[f(σ)] over all equiv-
ariant legal flows fon [T,σ] is attained (note that [T,σ] is not unimodular,
but the theorem can be used for P). Denote by f0the maximum flow. Let
wbe 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 wsatisfies 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 σof edge length r∈[bn, bn+1) in Rkcan be covered with at most
(b+ 1)kcubes of edge length bnin T0. If n≥0, the latter are either in Tor
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
constructed 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.
67
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 Φin Rkendowed 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 Frost-
man lemma. This lemma is also the basis of many results of [6].
7.4.1 Cayley Graphs
Proposition 7.18. For every finitely generated group Hwith 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
(Theorem 7.2) implies that for every M≥1, there exists w:D∗→R≥0such
that w(NM(e)) ≤Mαand E[w(e)] = ξα
M(H), where eis the neutral element
of H. Since the Cayley graph of His transitive and wis defined up to rooted
isomorphisms, w(H, ·) is constant. 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 cis 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 considering 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×D2is 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 arbi-
trary. By the unimodular Frostman lemma (Theorem 7.2), there is a nonnega-
tive measurable functions wion D∗such that ∀v∈Di:∀r≥1 : wi(Nr(v)) ≤
68
rα, a.s. In addition, wi(oi)6= 0 with positive probability. Consider the equiv-
ariant weight function won D1×D2defined by w(v1, v2) := w1[D1, v1]×
w2[D2, v2].It is left to the reader to show that wis an equivariant weight func-
tion. 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 Rof D1×D2
such that ER(o1,o2)α+β< . One has decay (λr(D2)) > β, where λris
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 D2with 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 R1of D1such
that ER1(o1)β< and ∀v∈D1:R1(v)∈ {0} ∪ [M, ∞) a.s. Choose the
extra randomness in R1independently from [D2,o2]. Given a realization of
[D1,o1] and R1, do the following: Let v1∈D1such that R1(v1)6= 0 (and
hence, R1(v1)≥M). One can find an equivariant subset Sv1of D2that gives a
covering of D2by 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,
0 otherwise.
Now, Ris a covering of D1×D2and it can be seen that it is an equivariant
covering. Also, given [D1,o1] and R1, the probability that o2∈So1is less than
R1(o1)−β. So one gets
ER(o1,o2)α+β=E"EhR(o1,o2)α+β|[D1,o1],R1i#
<ER1(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.
69
Example 7.21. Let Jbe a subset of Z≥0such that d(J) = 1 and d(J) = 0
simultaneously (see Subsection 6.5.1 for the definitions). Let Ψ1and Ψ2be
defined as in Subsection 6.5.1 corresponding to Jand Z≥0\Jrespectively.
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 Zas 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 embedding 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
embedding 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) = 0 or
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 Dexcept
in the trivial case where m(·) = 1 a.s.
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 em-
beddable in [Zk,0].
70
(iii) Let Hbe 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≥0such that E[w(o)] = ξα
M(D,1), and almost surely,
w(Nr(o)) ≤rαfor 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 w0on D0such 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)) ≤rαa.s. Therefore, the mass distribution prin-
ciple (Theorem 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 Conjec-
ture 8.4 is to consider an arbitrary equivariant covering of D0and try to extend
it to an equivariant covering of Dby adding some balls (without adding a ball
centered at the root). More generally, given an equivariant processes Z0on D0,
one might try to extend it to an equivariant process on Dwithout 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).
71
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 L1and 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 (Theo-
rem 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 connections to unimodular dimensions.
For subsets of Zd, the notions of upper and lower mass dimension are de-
fined 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⊆Rdand
calls it the macroscopic Minkowski dimension of A(one may define lower macro-
scopic Minkowski dimension similarly). This extension is obtained by pixelizing
Ato 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 unimod-
ular 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 ⊆Zdby large balls and considering the cost ( r
r+|x|)αfor each ball
in the covering, where rand xare the radius and the center of the ball and αis
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).
72
The unimodular Hausdorff dimension can be connected to the classical Haus-
dorff 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 compari-
son 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 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 Func-
tion
There exist unimodular discrete spaces Din which the udimH(D)-dimensional
Hausdorff 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 increasing 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≥0that is increasing in
αand such that ∀α > β : limr→∞ ϕα(r)/ϕβ(r) = ∞, one can redefine the uni-
modular Hausdorff dimension by sup{α:Mϕα(D)=0}(see e.g., the next para-
graph). 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 di-
mension by considering ϕα(r) := eαr. It might be useful for studying unimod-
ular spaces with super-polynomial volume growth, which are more interesting
73
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 Xis the union of kdisjoint copies of 1
rX, then the
similarity dimension of Xis log k/ log r(see e.g., [14]). This definition can also
be used for some infinite discrete sets as well. For instance, Zdis a union of 2d
copies of 2Zd. So, it can be said that the similarity dimension of Zdis 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
dimensions 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 rtends 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 direction of certain inequalities discussed in the paper: when adopt-
ing these negative unimodular dimensions, (i) the classical and unimodular
Minkowski and Hausdorff dimensions would be ordered in the same way, i.e.,
udimH(D)≤udimM(D)≤udimM(D); (ii) an equivariant subset of a unimod-
ular set would have a unimodular Minkowski dimension smaller than or equal
to that of the set, and possibly strictly smaller (see Subsections 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 cov-
erings; (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.
74
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⊆Zdis connected to whether the sim-
ple random walk in Zdhits Ainfinitely 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 Ghits
Dinfinitely often a.s.
2. Dimension functions. Does there exist a unimodular discrete space with-
out 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 off-
spring variance.
3. Simple random walk. By analogy with the image of subordinator pro-
cesses (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 ris a dimension function for Ψ. To prove this,
one should strengthen the bound Crβlog log rin the proof of Proposition 4.9
and also construct a covering of the set which is better than that of Proposi-
tion 3.13. For the former, one may use Theorem 4 of [26] (it seems 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 neces-
sary to use intervals with different lengths).
Another guess is that the image of the symmetric nearest-neighbor simple ran-
dom walk in Zdis 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 met-
ric (Subsection 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.
75
4. Eternal Galton-Watson trees. For unimodular eternal Galton-Watson
trees (Subsection 4.2.3), a conjecture is that if the offspring distribution is in
the domain of attraction 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 dimension 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 ex-
tended to a model in Zkfor k > 2 and 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 pro-
cess. 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).
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.
76
Acknowledgements
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 Euro-
pean Union’s Horizon 2020 research and innovation programme, grant agree-
ment 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.
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