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Unimodular Billingsley and Frostman Lemmas
Fran¸cois Baccelli∗
, Mir-Omid Haji-Mirsadeghi†
, and Ali Khezeli‡
February 3, 2020
Abstract
The notions of unimodular Minkowski and Hausdorff dimensions are
defined in [5] for unimodular random discrete metric spaces. The present
paper is focused on the connections between these notions and the polyno-
mial growth rate of the underlying space. It is shown that bounding the
dimension is closely related to finding suitable equivariant weight func-
tions (i.e., measures) on the underlying discrete space. The main results
are unimodular versions of the mass distribution principle, Billingsley’s
lemma and Frostman’s lemma, which allow one to derive upper bounds
on the unimodular Hausdorff dimension from the growth rate of suit-
able equivariant weight functions. These results allow one to compute or
bound both types of unimodular dimensions in a large set of examples
in the theory of point processes, unimodular random graphs, and self-
similarity. Further results of independent interest are also presented, like
a version of the max-flow min-cut theorem for unimodular one-ended trees
and a weak form of pointwise ergodic theorems for all unimodular discrete
spaces.
1 Introduction
This paper is a companion to [5]. For short, the latter will referred to as Part
I. The present paper uses the definitions and symbols of Part I. For cross refer-
encing the definitions and results of Part I, the prefix ‘I.’ is used. For example,
Definition I.3.16 refers to Definition 3.16 in Part I.
Part I introduced the notion of unimodular random discrete metric space
and two notions of dimension for such spaces, namely the unimodular Minkow-
ski dimension (Section I.3.1) and the unimodular Hausdorff dimension (Section
I.3.3).
The present paper is centered on the connections between these dimensions
and the growth rate of the space, which is the polynomial growth rate of #Nr(o),
where Nr(o) represents the closed ball of radius rcentered at the origin and
∗The University of Texas at Austin, baccelli@math.utexas.edu
†Sharif University of Technology, mirsadeghi@sharif.ir
‡Tarbiat Modares University, khezeli@modares.ac.ir
1
#Nr(o) is the number of points in this ball. Section 2 is focused on the basic
properties of these connections. It is first shown that the upper and lower poly-
nomial growth rates of #Nr(o) (i.e., limsup and liminf of log(#Nr(o))/log ras
r→ ∞) provide upper and lower bound for the unimodular Hausdorff dimen-
sion, respectively. This is a discrete analogue of Billingsley’s lemma (see e.g.,
[8]). A discrete analogue of the mass distribution principle is also provided,
which is useful to derive upper bounds on the unimodular Hausdorff dimension.
In the Euclidean case (i.e., for point-stationary point processes equipped with
the Euclidean metric), it is shown that the unimodular Minkowski dimension is
bounded from above by the polynomial decay rate of E[1/#Nn(o)]. Weighted
versions of these inequalities, where a weight is assigned to each point, are also
presented. As a corollary, a weak form of Birkhoff’s pointwise ergodic theorem
is established for all unimodular discrete spaces.
The bounds derived in Section 2 are fundamental for calculating the unimod-
ular dimensions. These bounds are used in Section 3 to complete the examples
of Part I. Some new examples are also presented in Section 3 for further illus-
tration of the results.
Section 4 gives a unimodular analogue of Frostman’s lemma. Roughly speak-
ing, this result states that there exists a weight function on the points such that
the upper bound in the mass distribution principle is sharp. This result is a
powerful tool to study the unimodular Hausdorff dimension and is the basis of
many of the results of [4]; e.g., connections to scaling limits, discrete dimension
and capacity dimension. In the Euclidean case, another proof of the unimod-
ular Frostman lemma is provided using a unimodular version of the max-flow
min-cut theorem, which is of independent interest.
It should be noted that the results regarding upper bounds on the dimen-
sion (e.g., in the unimodular mass distribution principle, Billingsley lemma and
Frostman lemma) are valid for arbitrary gauge functions as well (see Subsec-
tion I.3.8.2). The lower bounds are also valid for gauge functions satisfying the
doubling condition. However, these more general cases are skipped for simplicity
of reading.
2 Connections to Growth Rate
Let Dbe a discrete space and o∈D. The upper and lower (polynomial)
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)) <∞. If the upper and lower
growth rates are equal, the common value is called the 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.
2
In various situations in this paper, some weight in R≥0can be assigned to
each point of D. In these cases, it is natural to redefine the 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 equiv-
ariant processes of Subsection I.2.5. 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 I.2.5 for more on the matter). Also, given D, the weights are
allowed to be random.
Definition 2.1. An equivariant weight function wis an equivariant process
(Definition I.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 I.2.4). Lemma I.2.11
shows that [D,o;wD] is a random pointed marked discrete space and is uni-
modular.
In the following, ‘wD(·)is non-degenerate (i.e., not identical to zero) with
positive probability’ means that P[∃v∈D:wD(v)6= 0] >0.If [D,o] is unimod-
ular, Lemma I.2.14 implies that the above condition is equivalent to E[w(o)] >
0.Also, ‘wD(·)is non-degenerate a.s.’ means P[∃v∈D:wD(v)6= 0] = 1.
2.1 Unimodular Mass Distribution Principle
Theorem 2.2 (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 (I.3.3)
satisfies Hα
M(D)≥1
cE[w(o)] .
(ii) If in addition, wD(·)is non-degenerate with 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))] .(2.1)
Consider the independent coupling of wand R; i.e., for each deterministic dis-
crete space G, choose wGand RGindependently (see Definition I.2.8). Then, it
can be seen that the pair (w,R) is an equivariant process. So by Lemma I.2.11,
[G,o; (w,R)] is unimodular. Now, the mass transport principle (I.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.,
3
where the last inequality follows from the fact that Ris a covering. There-
fore, the mass transport 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 (2.1),
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, with positive probability, wD(·) is non-degenerate, then Lemma I.2.14
implies that w(o)>0 with positive probability. So E[w(o)] >0. Therefore,
Hα
1(D)>0 and the second claim is proved.
2.2 Unimodular Billingsley Lemma
The main result of this subsection is Theorem 2.6. It is based on Lemmas 2.3
and 2.4 below. Lemma 2.3 is a stronger version of the mass distribution principle
(Theorem 2.2).
Lemma 2.3 (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(·)is non-degenerate with 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.(2.2)
Let Rbe an equivariant covering such that R(·)∈ {0} ∪ [m, ∞) a.s. One
has
E[R(o)α]≥ER(o)α1{NR(o)∩Am6=∅}.(2.3)
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
NR(o)(o)⊆N2R(o)(v), it follows that w(NR(o)) ≤w(N2R(o)(v)) ≤2αc0R(o)α.
Therefore, (2.3) gives
E[R(o)α]≥1
2αc0Ew(NR(o))1{NR(o)∩Am6=∅}.(2.4)
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 princi-
ple (I.2.2) and (2.4) 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 (2.2) 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 also proved by arguments similar to those in Theorem 2.2.
4
Lemma 2.4 (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 inf r→∞ 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],
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.
5
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 2.5. The assumption in part (iii) of Lemma 2.4 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 2.6 (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))]) .
The proof is given below. In fact, in many examples, it is enough to consider
w≡1 in Billingsley’s lemma; i.e., w(Nr(o)) = #Nr(o).
Corollary 2.7. Under the assumptions of Theorem 2.6, if the upper and lower
growth rates of Dare almost surely constant (e.g., when [D,o]is ergodic), then,
growth (w(Nr(o))) ≤udimH(D)≤growth (w(Nr(o))) a.s. (2.5)
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 (2.5)
is valid for the sample Hausdorff dimension of D, which will be studied in [4].
6
Proof of Theorem 2.6. The first inequality is implied by part (ii) of Lemma 2.4.
The last inequality is implied by Lemma A.1. 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 2.3
implies that udimH(D)≤α. This proves the result.
Remark 2.8. In fact, the assumption E[w(o)] <∞in Theorem 2.6 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 I.2.6). 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 3.3.1.
Also, the condition that wDis non-degenerate a.s. is trivially necessary for the
upper bound.
Corollary 2.9. 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 2.4 implies the first claim. The second claim is implied by
Example I.3.17 (this can be deduced from the unimodular Billingsley lemma as
well).
Corollary 2.10. The unimodular Minkowski and Hausdorff dimensions of any
unimodular two-ended tree are equal to one.
This result has already been shown in Theorem I.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 2.11. In the setting of Corollary 2.7, 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 2.19
and the unimodular Frostman lemma (Theorem 4.2) below. Note that there are
examples where growth (·)6= growth (·) as shown in Subsections 3.2.1 and 3.5.1.
2.3 Bounds for Point Processes
The next results use the following equivariant covering. 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
7
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 order). Let R=Rϕ
assign 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 I.3.9).
Theorem 2.12 (Minkowski Dimension in the Euclidean Case).Let Φbe a
point-stationary point process in Rkand assume the metric in 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 I.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 I.3.10 implies both equalities in
the 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.
An example where the decay rate of E[1/#Nn(o)] is strictly smaller than
the growth rate of E[#Nn(o)] can be found in [4].
Proposition 2.13. 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 2.6) implies that udimH(Φ) ≤k.
Proposition 2.14. 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 measure of Ψ0satisfies M0
k(Ψ0)=2kρ(Ψ),
where ρ(Ψ) is the intensity of Ψ.
8
Notice that if Ψ0⊆Zk, then the claim is directly implied by Theorem I.3.34.
The general case is treated below.
Proof. For the first claim, by Proposition 2.13, it is enough to prove that
udimM(Φ) ≥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 Φ00 of Ψ00 is obtained by the superposition of Φ and an indepen-
dent stationary lattice with probability p(heads), and the superposition of Zk
and Ψ with probability 1−p(tails). So Lemma 2.4 implies that udimM(Φ00)≥k.
Note that Φ00 has two natural equivariant subsets which, after conditioning to
contain the origin, have the same distributions as Φ and Zkrespectively. There-
fore, one can use Theorem I.3.34 to deduce that udimM(Φ) ≥udimM(Φ00) = k.
Therefore, Proposition 2.13 implies that udimH(Φ) = udimM(Φ) = k.
Also, by using Theorem I.3.34 twice, one gets M0
k(Φ) = pM0
k(Φ00) and
M0
k(Zk) = (1 −p)M0
k(Φ00). Therefore, M0
k(Φ) = 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 I.3.29). This implies the claim.
The last claim of Proposition 2.14 suggests the following, which is verified
when k= 1 in the next proposition.
Conjecture 2.15. If Φis a point-stationary point process in Rkwhich is not
the Palm version of any stationary point process, then Mk(Φ) = 0.
Proposition 2.16. Conjecture 2.15 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 I.2.6). The assumption that Φ is not the Palm version of
a stationary point process is equivalent to E[S1] = ∞(see [10] or Proposition 6
of [16]). Indeed, if E[S1]<∞, then one could bias the probability measure by
S1(Definition I.C.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 [19] implies that
limn(T1+··· +Tn)/n =∞. This in turn implies that limr#Nr(0)/r = 0.
Therefore, Lemma 2.3 gives that H1
∞(Φ) = ∞; i.e., M1(Φ) = 0.
2.4 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 2.17. Let [D,o]be a unimodular discrete space. For any two equiv-
ariant weight functions w1and w2, if E[w1(o)] <∞and w2(·)is non-degenerate
a.s., then
growth (w1(Nr(o))) ≤growth (w2(Nr(o))) , a.s.
9
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 2.8.
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∈
D: [D, v]∈A), by a direct verification of the mass transport principle (I.2.1),
one can show that [D0,o0] is unimodular. So by using the unimodular Billingsley
lemma (Theorem 2.6) 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 2.18. Theorem 2.17 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 2.17. Note that Theorem 2.17 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 [5] that, roughly speaking,
non-amenability implies growth (w2(Nr(o))) = ∞, which makes the claim of
Theorem 2.17 trivial in this case. In this case, using exponential gauge func-
tions seems more interesting.
Problem 2.19. 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)] <∞?
2.5 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.,
[8]) 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., [8]). Theorem 2.6 does not require such assumptions. Note also that the
term growth (w(Nr(o))) in Theorem 2.6 does not depend on the origin in con-
trast to the analogous term in the continuum version. Similar observations can
be made on Theorem 2.2.
10
3 Examples
This section presents some examples for illustrating the results of the previous
section. It also provides further results on the examples introduced in Sec-
tion I.4.
3.1 General Unimodular Trees
The following is a direct corollary of Theorem I.4.2 and the unimodular Billings-
ley lemma. Since the statement does not involve dimension, it is of independent
interest and believed to be new.
Corollary 3.1. 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 3.2. Let [T,o]be a unimodular tree with infinitely many ends such
that E[deg(o)] <∞. Then Thas exponential growth a.s. and udimH(T) = ∞.
In fact, the assumption E[deg(o)] <∞is not necessary. Also, the graph-
distance metric on Tcan be replaced by an arbitrary equivariant metric. These
will be proved in [5].
The following proof uses the definitions and results of [1], but they are not
recalled for brevity.
Proof of Proposition 3.2. By Corollary 8.10 of [1], [T,o] is non-amenable (this
will be discussed further in [5]). So Theorem 8.9 of [1] 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 growth. Finally, the unimodular Billingsley lemma (Theorem 2.6)
implies that udimH(T) = ∞.
The following example shows that the Minkowski dimension can be finite.
Example 3.3. 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 [1] or [16]). 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
11
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 3.4 below
implies that udimM(T) = decay (P[le≥r]).
The following proposition gives a lower bound on the Minkowski dimension.
Proposition 3.4. 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 3.2.4 as well.
Proposition 3.5. 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 I.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 (this is implied by
Lemma A.3). Also, the assumption implies that E[w0(o)] <∞. So Lemma 2.4
implies that udimM(T3,d0)≥αand the claim is proved.
Proof of Proposition 3.4. 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 3.5, there exists c=c(α), such that
w0(Nr(v)) ≥crαfor all v∈Tand r≥0 (this is implied by Lemma A.3) and
the claim is proved.
3.2 Instances of Unimodular Trees
3.2.1 A Unimodular Tree With No Growth Rate
Recall the generalized canopy tree [T,o] from Subsection I.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
12
exist. In the latter, the existence of such trees is proved in [22], 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) growth rate. This proves the claim. Note also that
the unimodular Billingsley lemma and Theorem I.4.2 imply that udimH(T) =
growth (T) here.
3.2.2 Unimodular Galton-Watson Trees
Here, it is shown that the unimodular Galton-Watson tree [1] is infinite di-
mensional. (note that this tree differs from the Eternal Galton-Watson tree
of Subsection I.4.2.3 which is a directed tree). Consider an ordinary Galton-
Watson tree with offspring distribution µ= (p0, p1, . . .), where µis a probabil-
ity 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 3.6. 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 is followed from the unimodular
Billingsley lemma (Theorem 2.6) and the Kesten-Stigum theorem [15], 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 2.4, it is enough to prove
that E(1 −n−α)#Nn(o)has infinite decay rate for every α≥0. Denote by
[e
T,e
o] the Galton-Watson tree with the same parameters. Using the fact that
#Nn(o) is stochastically larger than #Nn−1(e
o), one gets that it is enough to
prove the last claim for [e
T,e
o].
For simplicity, the proof is given for the case p0= 0 only. By this assumption,
the probability of extinction is zero. The general case can be proved with similar
arguments and by using the decomposition theorem of supercritical Galton-
Watson trees (see e.g., Theorem 5.28 of [17]). 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.
13
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., [3]). 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 decayf(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.
3.2.3 Unimodular Eternal Galton-Watson Trees
Unimodular eternal Galton-Watson (EGW) trees were introduced in Subsection I.4.2.3.
The following theorem complements Theorem I.4.9.
Proposition 3.7. Let [T,o]be a unimodular eternal Galton-Watson tree. If
the offspring distribution has finite variance, then udimM(T) = udimH(T)=2.
Proof. Theorem I.4.9 proves that udimM(T) = 2. So it remains to prove
udimH(T)≤2. By the unimodular Billingsley lemma (Theorem 2.6), it is
enough to show that E[#Nn(o)] ≤cn2for a constant c. Recall from Subsec-
tion I.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 [6], Ynis a critical Galton-Watson
tree up to generation n. Also, for 0 ≤i<n,Yihas the same structure up to gen-
eration i, except that the distribution of the first generation is size-biased minus
one (i.e., (npn+1)nwith the notation of Subsection 3.2.2). So the assumption of
finite variance implies that the first generation in each Yihas finite mean, namely
m0. Now, one can inductively show that E[#Yn] = nand E[#Yi] = im0, for
0≤i<n. It follows that E[#Nn(o)] ≤(1 + m0)n2and the claim is proved.
3.2.4 The Poisson Weighted Infinite Tree
The Poisson Weighted Infinite Tree (PWIT) is defined as follows (see e.g., [2]). 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,
14
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 [2] for more details.
Proposition 3.8. 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 3.5 implies that udimM(T) = ∞(see also Lemma A.3). This proves
the claim.
3.3 Examples associated with Random Walks
As in Subsection I.4.3, consider the simple random walk (Sn)n∈Zin Rk, where
S0= 0 and the increments Sn−Sn−1are i.i.d.
3.3.1 The Image and The Zeros of the Simple Random Walk
Recall that Theorem I.4.11 studies the unimodular Minkowski dimension of the
image of a simple random walk. The following is a complement to this result.
Theorem 3.9. Let Φ := {Sn}n∈Zbe the image of a simple random walk Sin
R, where S0:= 0. Assume the jumps Sn−Sn−1are positive a.s.
(i) udimH(Φ) ≤1∧decay (P[S1> r]).
(ii) If β:= decay (P[S1> r]) exists, then udimM(Φ) = udimH(Φ) = 1 ∧β.
Proof. Theorem I.4.11 proves that udimM(Φ) ≥1∧decay (P[S1> r]). So it is
enough to prove part (i). Since Φ is a point-stationary point process in R(see
Subsection I.4.3.1), Proposition 2.13 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. By using Lemma A.2 twice, for the positive and negative parts of the
random walk, one can prove that there exists C < ∞and a random number
r0>0 such that for all r≥r0, one has #Nr(o)≤Crβlog log ra.s. Therefore,
the unimodular Billingsley lemma (Theorem 2.6) implies that udimH(Φ) ≤β+
for every > 0, which in turn implies that udimH(Φ) ≤β. So the claim is
proved.
The following proposition complements Theorem I.4.12. It is readily implied
by Theorem 3.9 above.
Proposition 3.10. Let Ψbe the zero set of the symmetric simple random walk
on Zwith uniform jumps in {±1}. Then, udimM(Ψ) = udimH(Ψ) = 1
2.
In this proposition, the guess is that M1/2(Ψ) = ∞. Additionally, by anal-
ogy with the zero set of Brownian motion [21], it is natural to guess that
√rlog log ris a dimension function for Ψ (see Subsection I.3.8.2). To prove
15
this, one should strengthen Lemma A.2 and also construct a covering of the
set which is better than that of Proposition I.3.14. For the former, one may
use Theorem 4 of [12] (it seems that the assumption of [12] on the tail of the
jumps is not necessary for having an inequality similarly to Lemma A.2). For
the latter, one might try to get ideas from [21] (it is necessary to use intervals
with different lengths).
Example 3.11 (Infinite Hausdorff Measure).In Theorem 3.9, assume P[S1> r] =
1/log rfor large enough r. Then, part (ii) of the theorem implies that udimH(Φ) =
0. However, since Φ is infinite a.s., it has infinite 0-dimensional Hausdorff mea-
sure (Proposition I.3.28).
Example 3.12 (Zero Hausdorff Measure).In Theorem 3.9, assume P[S1> r] =
1/r for large enough r. Then, part (ii) of the theorem implies that udimH(Φ) =
1. Since E[S1] = ∞, Φ is not the Palm version of any stationary point process
(see Proposition 2.16). Therefore, Proposition 2.16 implies that M1(Φ) = 0.
3.3.2 The Graph of the Simple Random Walk
The graph of the random walk (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 I.4.3.1). Since #Ψ∩[−n, n]k+1 ≤2n+ 1, the mass
distribution principle (Theorem 2.2) 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. There-
fore, the unimodular Billingsley lemma (Theorem 2.6) implies that udimH(Ψ) ≥
1. Hence, udimH(Ψ) = 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|}.(3.1)
Theorem I.3.31 implies that, by considering this metric, unimodularity is
preserved and dimension is not decreased. Under this metric, the ball Nn(0) is
Ψ∩[−n2, n2]×[−n, n]. It is straightforward that Z2has growth rate 3 and also
Minkowski and Hausdorff dimension 3 under this metric.
Proposition 3.13. If the jumps are ±1uniformly, under the metric (3.1), 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 2.2) 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 I.3.12). 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
16
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.
3.4 A Drainage Network Model
Let [T,o] be the one-ended tree in Subsection I.4.5 equipped with the graph-
distance metric.
Proposition 3.14. One has udimM(T) = udimH(T) = 3
2.
Proof. Theorem I.4.14 proves that udimM(T) = 3
2. So it is enough to prove
udimH(T)≤3
2. To use the unimodular 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 #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 2.6) implies that udimH(T)≤
3
2. So the claim is proved.
3.5 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 I.4.6.
3.5.1 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 I.4.6, one can shift this set randomly and take a limit
to obtain a unimodular discrete space. This can be constructed in the following
way as well: Let T0:= {0}. If n∈J, let Tn+1 := Tn∪(Tn±2×2n), where the
sign is chosen i.i.d., each sign with probability 1/2. If n6∈ J, let Tn+1 := Tn.
Finally, let Ψ := ∪nTn.
The upper and lower asymptotic densities of Jin Z≥0are defined by d(J) :=
lim supn1
nJnand d(J) := lim inf n1
nJn, where Jn:= #J∩ {0, . . . , n}.
17
Proposition 3.15. 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) growth rate does not exist.
Proof. Let n≥0 be given. Cover Tnby a ball of radius 2ncentered at the min-
imal element of Tn. By the same recursive definition, one can cover Tn+1 by ei-
ther 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 I.3.9), Lemma I.3.10
implies that udimM(Ψ) = d(J) and udimM(Ψ) = d(J). One has
#Tm= 2Jm.(3.2)
This implies that #N2n(o)≤2Jn+1. One can deduce that growth(#Nn(o)) ≤
d(J). So the unimodular Billingsley lemma (Theorem 2.6) gives udimH(Ψ) ≤
d(J). This proves the claim.
3.5.2 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[14] (see also [8]) 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 [8] 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 3.16. Let K0
ndenote the random set obtained by biasing the dis-
tribution of Knby #Kn(Definition I.C.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].
(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.
18
Note that in contrast to the continuum analogue [14], 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 3.17. 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 3.16.
Proof of Lemma 3.17. 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 I.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 3.16. 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
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.
19
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 2.6) 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 2.12. This covering satisfies P[Rn(o)≥0] = Eh1/#(Cn∩ˆ
K)i.Let
[T0,o0] be the eternal Galton-Watson tree of Subsection I.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 [6] for more details on eternal
family trees). Therefore, Lemma 5.7 of [6] 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 [6] implies that the generation of the root is infinite a.s. This proves the
claim.
3.6 Cayley Graphs
As mentioned in Subsection I.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 growth rate of the group. Note that Gromov’s theorem [13] implies that the
polynomial growth degree exists and is either an integer or infinity.
Theorem 3.18. For every finitely generated group Hwith polynomial growth
degree α∈[0,∞], one has udimM(H) = udimH(H) = αand Mα(H)<∞.
Proof. First, assume α < ∞. The result of Bass [7] implies that there are
constants c, C > 0 such that ∀r≥1 : crα<#Nr(o)≤C rα, where ois an
arbitrary element of H. So the mass distribution principle (Theorem 2.2) and
part (i) of Lemma 2.4 imply that udimM(H) = udimH(H) = α. In addition,
(2.1) in the proof of Theorem 2.2 implies that Hα
M(H)≥1/C for all M≥1,
which implies that Mα(H)≤C < ∞.
Second, assume α=∞. The result of [23] shows that for any β < ∞,
#Nr(o)> rβfor sufficiently large r. Therefore, part (i) of Lemma 2.4 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 4.17.
20
3.7 Notes and Bibliographical Comments
The proof of Proposition 3.2 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 I.4. The example defined by digit restriction (Subsection 3.5.1)
is inspired by an example in the continuum setting (see e.g., Examples 1.3.2
of [8]). The randomized discrete Cantor set (Subsection 3.5.2) is inspired by the
random cantor set (see e.g., Section 3.7 of [8]).
4 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 4.4.2 and 4.4.3). It is also the basis of many of the results in [5].
4.1 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 4.8 below
defines a metric on Ξ such that the notion of Ξ-valued equivariant processes
(Definition I.2.8) are well defined. Such a process cis called an equivariant
weighted collection of balls1. 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.
Definition 4.1. 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., (4.1)
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).
1The term ‘weighted’ refers to the weighted sums in Definition 4.1 and should not be
confused with equivariant weight functions of Definition 2.1
21
It is straightforward that every equivariant ball-covering of Definition I.3.15
gives a (1,1)-covering, where 1 is regarded as the constant function f≡1 on
D∗. This gives (see also Conjecture 4.4 below)
ξα
M(1) ≤ Hα
M(D).(4.2)
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 following theorem, to be consistent with the setting of the paper, the
following notation is used: w(u) := w([D, u]) for u∈D, and w(Nr(v)) =
Pu∈Nr(v)w(u). Also, recall that a deterministic equivariant weight function is
given by a measurable function w:D∗→R≥0(see Example I.2.10).
Theorem 4.2 (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.(4.3)
(ii) Given a non-negative function h∈L1(D∗, µ), the first condition can be
replaced by E[w(o)h(o)] = ξα
M(h).
(iii) In the setting of (ii), if Dhas finite α-dimensional Hausdorff measure
(e.g., when udimH(D)> α) and h6≡ 0, then w[D,o]6= 0 with positive
probability.
The proof is given later in this subsection.
Remark 4.3. One can show that (4.3) implies that 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 (4.3) (it should be noted that the maximal is not unique). The proof
is similar to that of the mass distribution principle (Theorem 2.2) and is left to
the reader.
Conjecture 4.4. One has Hα
M(D) = ξα
M(1).
Lemma 4.7 below proves a weaker inequality. For instance, it can be seen
that the conjecture holds for Zk(with the l∞metric) and for the non-ergodic
example of Example I.3.20. In the former, 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 is similar by letting w[Z,0] := M
2M+1
and w[Z2,0] := 0.
22
Remark 4.5. The unimodular Frostman lemma implies that, in theory, the
mass transport principle (Theorem 2.2) 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 (4.3) can be found; e.g.,
for two-ended trees). Therefore, in practice, the unimodular Billingsley lemma
is more useful than the mass transport principle.
The following lemma is needed to prove Theorem 4.2.
Lemma 4.6. 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 (4.1) 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 (4.1) 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 cwas arbitrary, one obtains ξα
M(f2)≤ξα
M(f1) + MαE[|f2(o)−f1(o)|],
which implies the claim.
Proof of Theorem 4.2. Part (i) is implied by Part (ii) which is proved below. 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 [20]), there is a linear functional l:L1(D∗, µ)→R
such that l(h) = ξα
M(h) and −ξα
M(−f)≤l(f)≤ξα
M(f),for all f∈L1. Since l
is sandwiched between two functions which are continuous at 0 and are equal
at 0, 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 l(f) = E[f(o)w(o)] ,for all f∈L1. Note
that if f≥0, then ξα
M(−f) = 0 and so l(f)≥0. This implies that w(o)≥0 a.s.
(otherwise, let f(o) := 1{w(o)<0}to get a contradiction). Consider a version of w
which is nonnegative everywhere. The claim is that wsatisfies the requirements.
Let r≥Mbe fixed. For all discrete spaces D, let S:= SD:= {v∈D:
w(Nr(v)) > rα}. Define fr(v) := #Nr(v)∩S. By the definition of SD, one has
Ew(Nr(o))1{o∈S}≥rαP[o∈S].(4.4)
Moreover, if P[o∈S]>0, then the inequality is strict. On the other hand,
by the mass transport principle for the function (v, u)7→ w(u)1{v∈S}1{u∈Nr(v)},
23
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 (4.1) for fr, which implies the last inequality by the definition of
ξα
M(·). Thus, equality holds in (4.4). Hence, P[o∈S] = 0; i.e., w(Nr(o)) ≤rα
a.s. Lemma I.2.14 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 desired.
Also, one has E[w(o)h(o)] = l(h) = ξα
M(h).Thus, wsatisfies the desired
requirements.
To prove (iii), assume h6≡ 0 and Mα(D)<∞. By Lemma I.3.25, one has
Hα
M(D)>0. So Lemma 4.7 below implies that ξα
M(h)>0. Now, the above
equation implies that wis not identical to zero.
The above proof uses the following lemma.
Lemma 4.7. 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 06=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.
(iii) In addition, ξα
M(h)=0if and only if Hα
M(D)=0.
Proof. (i). By considering the cases where c(·)∈ {0,1}, the first inequality
is easily obtained from the definition of ξα
1(1). In particular, this implies that
b≤1. The second inequality is implied by part (ii) by letting h(·) := 1 and
a:= −log b≥0.
(ii). Let b0> ξα
M(h) be arbitrary. So there exists an equivariant weighted
collection of balls cthat satisfies (4.1) 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
24
gives an equivariant covering, namely R, by balls of radii at least M. Then, one
can easily get
E[R(o)]α≤MαP[o6∈ S] + E
X
r≥
(acr(o)∧1)rα
≤MαP[o6∈ S] + ab0.(4.5)
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 (4.1), 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, (4.5) 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→ ∞ and using the first claim,
one obtains that Hα
M(D) = 0. Conversely, assume Hα
M(D) = 0. The first
inequality in part (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 4.8. 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., [9]). So
one can define the notion of Ξ0-valued equivariant processes as in Definition I.2.8.
Therefore, Ξ-valued equivariant processes also make sense.
4.2 Max-Flow Min-Cut Theorem for Unimodular One-
Ended Trees
The result of this subsection is used in the next subsection for a Euclidean
version of the unimodular Frostman lemma, but is of independent interest as
well.
The max-flow min-cut theorem is a celebrated result in the field of graph
theory (see e.g., [11]). 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.,
[8]).
25
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., [6]), 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 I.4.1.2, let F(v) be the
parent of vertex vand D(v) be the descendants subtree of v.
Definition 4.9. Alegal equivariant flow on [T;c] is an equivariant way
of assigning extra marks f(·)∈Rto the edges (see Definition I.2.8 and Re-
mark I.2.12), 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).(4.6)
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 (4.6) 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 4.11
below.
Below, since each edge of Tcan be uniquely represented as (v, F (v)), the
following convention is helpful.
Convention 4.10. 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 4.11. The norm of the legal equivariant flow fis defined as
|f|:= Ef(o)1{o∈L}.
26
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 (I.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 A.4).
Lemma 4.12. 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 2.2). 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 I.2.11). 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 v
to τ(v). By the mass transport principle (I.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 (see Lemma A.4) and the claim
follows.
The main result is the following converse to the above lemma.
Theorem 4.13 (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),
where the maximum is over all legal equivariant flows fand the infimum is over
all equivariant cut-sets Π.
27
Remark 4.14. The claim of Theorem 4.13 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 (I.2.2) holds. The same proof works for this case as well.
This will be used in Subsection 4.3.
Proof of Theorem 4.13. 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 (4.6) for vertices
of Tn\Lexcept 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 [11] (see
e.g., Theorem 3.1.5 of [8]) 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 (I.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 (the
claim is similar to Lemma I.B.3 and is left to the reader). Therefore, there is a
sequence n1, n2, . . . and an equivariant process f0such that fni→f0(weakly).
It is not hard to deduce that f0is a legal equivariant 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 4.12 implies that equality
holds and the claim is proved.
28
4.3 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 4.2.
As will be seen, the claim implies that in this case, Conjecture 4.4 holds up to a
constant factor (Corollary 4.16). However, the weight function obtained in the
theorem needs extra randomness.
Theorem 4.15. Let Φbe a point-stationary point process in Rkendowed with
the Euclidean 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(4.7)
and
E[w(0)] ≥3−kHα
1(Φ).(4.8)
In particular, if Hα
1(Φ) >0, then w(0) is not identical to zero.
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 2.3. 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 I.3.31).
Moreover, Theorem 5 of [16] 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)]
,(4.9)
29
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 4.14) 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 (4.9). Consider the conductance function c(τ) :=
bnα for all cubes τof edge length bnin Tand all n. Therefore, Theorem 4.13
and Remark 4.14 imply that the maximum of E[f(σ)] over all equivariant 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 (4.7) and (4.8).
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 (4.7) is
proved for w.
To prove (4.8), 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 n, 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 (4.9), 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 4.13) 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.
30
The following corollary shows that in the setting of Theorem 4.15, the claim
of Conjecture 4.4 holds up to a constant factor (compare this with Lemma 4.7).
Corollary 4.16. For all point-stationary point processes Φin Rkendowed with
the Euclidean metric and all α≥0,3−kHα
1(Φ) ≤ξα
1(Φ) ≤ Hα
1(Φ).
Proof. The claim is directly implied by (4.2), Theorem 4.15 and Remark 4.3.
4.4 Applications
The following are some basic applications of the unimodular Frostman lemma.
This lemma is also the basis of many results of [4].
4.4.1 Cayley Graphs
Proposition 4.17. For every finitely generated group Hwith polynomial growth
degree α∈[0,∞], one has ξα
∞(H)<∞.
Note that if Conjecture 4.4 holds, then this result implies Mα(H)>0, as
conjectured in Subsection 3.6.
Proof. By Theorem 3.18, Mα(H)<∞. So the unimodular Frostman lemma
(Theorem 4.2) 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 3.18. By letting M→ ∞, one gets ξα
∞(H)≤1/c < ∞.
4.4.2 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 [1]).
Proposition 4.18. 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).
(4.10)
Proof. By Theorem I.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 4.2), there is a nonnega-
tive measurable functions wion D∗such that ∀v∈Di:∀r≥1 : wi(Nr(v)) ≤
31
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 2.2) 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 (I.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 I.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 (4.10) are
strict.
Example 4.19. Assume [D1,o1] and [D2,o2] are unimodular discrete spaces
such that udimM(G1)<udimH(G1) and udimM(G2) = udimH(G2). By
Proposition 4.18, 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 (4.10) is obtained.
32
Example 4.20. Let Jbe a subset of Z≥0such that d(J) = 1 and d(J) = 0
simultaneously (see Subsection 3.5.1 for the definitions). Let Ψ1and Ψ2be
defined as in Subsection 3.5.1 corresponding to Jand Z≥0\Jrespectively.
Proposition 3.15 implies that udimH(Ψ1) = udimH(Ψ2) = 1. On the other
hand, (3.2) 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 2.6) implies that udimH(Ψ1×Ψ2)≤1 (in fact, equality holds by
Proposition 4.23 below). So the rightmost inequality in (4.10) is strict here.
4.4.3 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] as defined in Definition I.2.13. By the following definition,
[Z,0] is called embeddable in [Z2,0]. The dimension of embedded subspaces is
studied in this subsection, which happens to be non-trivial and requires the
unimodular Frostman lemma.
Definition 4.21. 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., (I.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 as above, [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 4.22. The following are instances of Definition 4.21.
(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].
33
(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 4.23. If [D0,o0]and [D,o]are unimodular discrete spaces and
the former is equivariantly embeddable in the latter, then
udimH(D)≥udimH(D0),(4.11)
ξα
M(D,1) ≤ Hα
M(D0),(4.12)
for all α≥0and M≥1, where ξα
Mis defined in Definition 4.1.
Proof. First, assume (4.12) holds. For α > udimH(D), one has Hα
M(D)>0
(Lemma I.3.25). Therefore, Lemma 4.7 implies that ξα
M(D,1) >0. Hence,
(4.12) implies that Hα
M(D0)>0, which implies that udimH(D0)≤α. So it is
enough to prove (4.12).
By the unimodular Frostman lemma (Theorem 4.2), 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 4.21. 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], Proposition I.B.1 gives an
equivariant process w0on D0such that [S,o0;w0] has the same distribution as
[D0,o0;w0]. 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 2.2) 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
measure. This is stated in the following conjecture.
Conjecture 4.24. Under the setting of Proposition 4.23, for all α > 0, one
has Mα(D)≥ Mα(D0).
Note that in the case α= 0, the conjecture is implied by Proposition I.3.28.
Also, in the general case, the conjecture is implied by (4.12) and Conjecture 4.4.
Another problem is the validity of Proposition 4.23 under the weaker assump-
tion of being non-equivariantly embeddable. As a partial answer, if growth (#Nr(o))
exists, then (4.11) 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),
34
where the first inequality and the last equality are implied by the unimodular
Billingsley lemma (Theorem 2.6).
Remark 4.25. Another possible way to prove Proposition 4.23 and Conjec-
ture 4.24 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).
4.5 Notes and Bibliographical Comments
The unimodular Frostman lemma (Theorem 4.2) is analogous to Frostman’s
lemma in the continuum setting (see e.g., Thm 8.17 of [18]). The proof of
Theorem 4.2 is also inspired by that of [18], but there are substantial differences.
For instance, the proof of Lemma 4.7 and also the use of the duality of L1and L∞
in the proof of Theorem 4.2 are new. The Euclidean version of the unimodular
Frostman’s lemma (Theorem 4.15) and its proof are inspired by the continuum
analogue (see e.g., [8]).
As already explained, the unimodular max-flow min-cut theorem (Theo-
rem 4.13) is inspired by the max-flow min-cut theorem for finite trees. Also,
the results and examples of Subsection 4.4.2 on product spaces are inspired by
analogous in the continuum setting; e.g., Theorem 3.2.1 of [8].
A Appendix
Lemma A.1. Let (Xn)∞
n=1 ≥0be a monotone sequence of random variables.
Then almost surely, growth (Xn)≤growth (E[Xn]).
One can also deduce that growth (Xn)≤growth (E[Xn]), but this is skipped
since it is not needed here.
Proof. The claim will be proved assuming 0 ≤X1≤X2≤ ··· . The non-
increasing case can be proved with minor changes. Let αand βbe arbitrary such
that growth (E[Xn]) < β < α. So there is a constant csuch that E[Xn]≤cnβ
for all n≥1. Let M:= max{n:Xn> nα}, with the convention max ∅:=
0. Below, it will be shown that M < ∞a.s. Assuming this, it follows that
growth (Xn)≤αa.s. By considering this for all αand β, the claim is implied.
Now, it is proved that M < ∞a.s. With an abuse of notation, the constant
35
cbelow is updated in each step without changing the symbol.
P[M≥n] = P[∃k≥n:Xk> kα]≤
∞
X
j=0
P∃k:n2j≤k≤n2j+1, Xk> kα
≤
∞
X
j=0
PXn2j+1 >(n2j)α≤
∞
X
j=0
E[Xn2j+1 ]
(n2j)α≤
∞
X
j=0
c(n2j+1)β
(n2j)α
≤
∞
X
j=0
c(n2j)β−α≤cnβ−α.
The RHS is arbitrarily small for large n. This implies that M < ∞a.s. and the
claim is proved.
Lemma A.2. Let X, X1, X2, . . . be a non-negative i.i.d. sequence and t > 0be
such that P[X > r]≥cr−tfor large enough r. Let Sn:= X1+· ·· +Xn. Then
there exists C < ∞such that almost surely,
∃n:∀k≥n:S−1(k)≤Cktlog log k.
Proof. First, one has
PS−1(n)≥m=P[Sm≤n]≤P[∀i≤m:Xi≤n] = P[X≤n]m
≤(1 −cn−t)m≤e−cmn−t.(A.1)
Let C:= 2t+1/c and ψ(x) := Cxtlog log x, Therefore, for large n, one has
P∃k≥n:S−1(k)> ψ(k)=Pmax
k≥n
S−1(k)
ψ(k)>1
≤
∞
X
j=0
Pmax
n2j≤k<n2j+1
S−1(k)
ψ(k)>1≤
∞
X
j=0
PS−1(n2j+1)> ψ(n2j)
≤
∞
X
j=0
e−cψ(n2j)(n2j+1)−t≤
∞
X
j=0
e−2 log log(n2j)=
∞
X
j=0
1
(jlog 2 + log n)2.
It is clear that the sum in the last term is convergent. Therefore, dominated
convergence implies that the right hand side tends to zero as n→0. This proves
the claim.
Lemma A.3. Let α < ∞and (T , o)be a deterministic rooted tree such that
deg(o)≥2and deg(v)≥3for all v6=o. Let d0be a metric on Twhich is
generated by edge lengths such that d0(·)≥1. Let w(u) := CPv∼ud0(u, v)α.
Then C=C(α)can be chosen such that for all r≥0, one has w(N0
r(o)) ≥rα,
where N0
rdenotes the ball of radius runder the metric d0.
Proof. Let Cbe a constant such that ∀x∈[0,1] : Cxα+ (1 −x)α≥1
2. It is easy
to see that such Cexists. For r≥0, let f(r) be the infimum value of w(N0
r(o))
36
for all trees with the stated conditions. So one should prove f(r)≥rα. The
claim is true for r= 0. Also, if 0 < r < 1, one has N0
r(o) = {o}and the
claim is trivial. The proof uses induction on brc. Assume that r≥1 and for
all s < brc, one has f(s)≥sα. For y∼o, let Tybe the connected component
containing ywhen the edge (o, y) is removed. It can be seen that [Ty, y ] satisfies
the conditions of the lemma. Therefore,
w(N0
r(o)) = w(o) + X
y:y∼o
w(N0
r−d0(o,y)(Ty, y)) ≥w(o) + X
y:y∼o
f(r−d0(o, y))
≥X
y:y∼o
[Cd0(o, y )α+ (r−d0(o, y))α]
≥deg(o)·min
0≤x≤r{Cxα+ (r−x)α} ≥ deg(o)rα/2≥rα,
where the third line is by the definition of w(o) and the induction hypothesis,
the fifth line is due to the definition of C, and the last line is by the assumption
deg(o)≥2. Hence f(r)≥rα, which proves the induction claim.
Lemma A.4. An equivariant cut-set is equivariantly minimal if and only if it
is almost surely minimal.
Proof. Let Π be an equivariant cut-set. If Π is almost surely minimal, then it is
also equivariantly minimal by definition. Conversely, assume Π is equivariantly
minimal but not almost surely minimal. Call an edge e0above an edge eif e0
separates efrom the end. Call an edge e∈Πbad if there is an edge of Π above
e. Let Π0be the set of bad edges of Π. Let Π00 be the set of lowest edges in Π0;
i.e., the edges e∈Π0such that there is no other edge of Π0below e. It can be
seen that the assumption implies that Π00 is nonempty with positive probability.
Now, it can be seen that Π \Π00 is an equivariant cut-set, which contradicts the
minimality of Π.
Acknowledgements
Supported in part by a grant of the Simons Foundation (#197982) to The
University of Texas at Austin.
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