On the Dimension of Unimodular Discrete Spaces
Part II: Relations with Growth Rate
The notions of unimodular Minkowski and Hausdorff dimensions are defined in  for unimodular random discrete metric spaces. The present paper is focused on the connections between these notions and the polynomial growth rate of the underlying space. It is shown that bounding the dimension is closely related to finding suitable equivariant weight functions (i.e., measures) on the underlying discrete space. The main results are unimodular versions of the mass distribution principle and Billingsley’s lemma, which allow one to derive upper bounds on the unimodular Hausdorff dimension from the growth rate of suitable equivariant weight functions. Also, a unimodular version of Frostman’s lemma is provided, which shows that the upper bound given by the unimodular Billingsley lemma is sharp. 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.
- 1 Introduction
- 2 Connections to Growth Rate
- 3.1 General Unimodular Trees
- 3.2 Instances of Unimodular Trees
- 3.3 Examples associated with Random Walks
- 3.4 A Drainage Network Model
- 3.5 Self Similar Unimodular Spaces
- 3.6 Cayley Graphs
- 4 Frostman’s Theory
- A Appendix
This paper is the second in a series of three (, the current paper, and ), which are referred to as Part I, II and III below. The present paper uses the definitions and symbols of Part I. A list of notation is provided in Table 1 at the end of the paper to ease the reading. For cross referencing the definitions and results of Part I, the prefix ‘I.’ is used. For example, Definition I.2.1 refers to Definition 2.1 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 Minkowski dimension (Section I.3.1 in Part I) 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 , where represents the closed ball of radius centered at the origin and 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 polynomial growth rates of (i.e., limsup and liminf of as ) provide upper and lower bound for the unimodular Hausdorff dimension, respectively. This is a discrete analogue of Billingsley’s lemma (see e.g., ). A discrete analogue of the mass distribution principle is then provided, which is useful to derive upper bounds on the unimodular Hausdorff dimension. In the particular case of a point-stationary point process equipped with the Euclidean metric, it is also shown that the unimodular Minkowski dimension is bounded from above by the polynomial decay rate of . Weighted versions of these inequalities, where a weight is assigned to each point, are also presented.
Section 3 is devoted to examples. It continues the main example section of Part II and introduces further examples. It is shown there that the bounds established in Section 2 are very useful for calculating the unimodular dimensions in the main instances of unimodular discrete spaces discussed in Part I, namely point processes, random graphs and self similar random discrete sets.
Section 4 gives a unimodular analogue of Frostman’s lemma. Roughly speaking, this lemma states that there is a weight function such that the upper bound in Billingsley’s lemma is sharp. This lemma provides a powerful tool to study the unimodular Hausdorff dimension, in particular, to assess the dimension of subspaces and product spaces. In the Euclidean case, another proof of the unimodular Frostman lemma is provided using a unimodular version of the max-flow min-cut theorem, which is of independent interest.
2 Connections to Growth Rate
Let be a discrete space and . The upper and lower (polynomial) growth rates of are
has finite polynomial growth if . If the upper and lower growth rates are equal, the common value is called the growth rate of .
For , one has and , where . This implies that and do not depend on the choice of the point .
In various situations in this paper, some weight in can be assigned to each point of . In these cases, it is natural to redefine the growth rate by considering the weights; i.e., by replacing with the sum of the weights of the points in . This will be formalized below using the notion of equivariant processes of Subsection I.2.6.
In the following definition, weights should be defined for all discrete spaces . However, if a random pointed discrete space is considered, it is enough to define weights in almost every realization (see Subsection I.2.6 for more on the matter). Also, given , the weights are allowed to be random.
An equivariant weight function is an equivariant process (Definition I.2.6) with values in . For a discrete space and , the (random) value is called the weight of . Also, for , let
The last equation shows that one could also call an equivariant measure.
Assume is a unimodular discrete space (Subsection I.2.5). Lemma I.2.28 shows that is a random pointed marked discrete space and is unimodular. Also, one can let be undefined for a class of discrete spaces, as long as is almost surely defined.
In the following, the term ‘ is non-degenerate (i.e., not identical to zero) with positive probability’ means that
In the case when is unimodular, Lemma I.2.30 implies that the above condition is equivalent to
Also, the term ‘ is non-degenerate a.s.’ means that
2.1 Unimodular Mass Distribution Principle
Theorem 2.2 (Mass Distribution Principle).
Let be an arbitrary equivariant covering such that a.s. By the assumption on , a.s. Therefore,
By letting , one gets . Also, a.s., where the last inequality follows from the fact that is a covering. Therefore, the mass transport principle I.2.3 implies that (recall that by convention, is the empty set when ). So by (2.3), one gets . Since this holds for any , one gets that and the first claim is proved.
If, with positive probability, is non-degenerate, then Lemma I.2.30 implies that with positive probability. So . Therefore, and the second claim is proved. ∎
Theorem 2.2, applied to the counting measure , implies that and . As already proved in Propositions I.3.24 and I.3.38, equality holds in both.
The assumption (2.2) in Theorem 2.2 implies a uniform polynomial bound
on the size of all balls by the fact that what holds a.s. at the root, holds a.s. at all points
In practice, this assumption is not as applicable as its continuum counterpart (Lemma 1.9 in ), except for some simple examples. The unimodular Billingsley lemma in the next subsection will actually be more useful. See also Lemma 2.5 below.
2.2 Unimodular Billingsley Lemma
Lemma 2.5 (An Upper Bound).
Let be a unimodular discrete space.
If and is an equivariant weight function such that
In addition, if is non-degenerate with positive probability (see (2.1)), then .
Let be arbitrary. The assumption implies that a.s. For , let
which is an increasing sequence of equivariant subsets. Since ,
Let be an equivariant covering such that a.s. One has
If , then and hence . In the next step, assume that this is the case. Let be an arbitrary point in . By the definition of , one gets that for all , Since , it follows that . Therefore, (2.5) gives
By letting , one gets that . Also, since there is a ball that covers a.s., one has a.s. Therefore, the mass transport principle (I.2.3) and (2.6) imply that
This implies that . Using (2.4) and letting tend to infinity gives . Since is arbitrary, the first claim is proved.
Lemma 2.6 (A Lower Bound).
Let be a unimodular discrete space, and . Let be an arbitrary equivariant weight function such that .
If a.s., then .
If a.s., then .
If , then .
If , then .
The proofs of the first two parts are very similar. So the second part is proved first.
(ii). Let , and be such that . Fix . Let be the equivariant subset obtained by selecting each point with probability independently. Let if , if , and otherwise. Then is an equivariant covering. It is shown below that .
Let . By the assumption, a.s. One has
Therefore, . It follows that . Since is arbitrary, this implies .
(i). Only a small change is needed in the above proof. For , let if either or , and let otherwise. Note that is a covering by balls of equal radii. By the same computations and the assumption , one gets
which is of order for large . This implies that . Since is arbitrary, one gets and the claim is proved.
(iii). Let be arbitrary. It will be proved below that there is a sequence such that . If so, by a slight modification of the proof of Part (ii), one can find a sequence of equivariant coverings such that and (iii) is proved.
Let be arbitrary. By the assumption, there is and such that . So
Note that for fixed and as above, can be arbitrarily large. Now, choose large enough for the right hand side to be at most . This shows that can be arbitrarily small and the claim is proved.
Theorem 2.8 (Unimodular Billingsley Lemma).
Let be a unimodular discrete metric space. For all equivariant weight functions such that
If the upper and lower growth rates of are almost surely constant (e.g., when is ergodic), then, almost surely,
The first part is implied by the second one. So it is enough to prove the second part.
Under the assumptions of Theorem 2.8, if exists and is constant a.s., then
Subsection 3.5.1 below provides an example where .
In fact, the assumption in Theorem 2.8
is only needed for the lower bound while the assumption is only needed for the upper bound.
These assumptions are also necessary as shown below.
For example, assume is a point-stationary point process in (see Example I.2.18). For , let be the sum of the distances of to its next and previous points in . This equivariant weight function satisfies for all , and hence . But can be strictly less than 1 as shown in Subsection 3.3.1.
Also, the condition that is non-degenerate a.s. is trivially necessary for the upper bound.
A converse to the unimodular Billingsley lemma is the unimodular Frostman lemma, which will be discussed in Section 4.
Here is a first application of the unimodular Billingsley lemma.
Let be a unimodular random graph equipped with the graph-distance metric. If is infinite almost surely, then and else, .
For all discrete spaces , let if is finite and otherwise. One has . If is finite with positive probability, then . Therefore, the unimodular Billingsley lemma (Theorem 2.8) implies that , which in turn implies the second claim. ∎
The upper bound in Theorem 2.8 is analogous to Billingsley’s lemma in the continuum setting (see e.g., Lemma 3.1 in ). It is interesting that there is no need to assume is embedded in or the bounded subcover property holds (see Remark 3.2 in ). Note also that does not depend on the origin in contrast to the analogous term in the continuum version.
2.3 Bounds for Point Processes
The next results use the following equivariant covering. Let be a discrete subset of equipped with the metric and . Let , be a point chosen uniformly at random in , and consider the partition of by cubes. Then, for each , choose a random element in independently (if the intersection is nonempty). The distribution of this random element should depend on the set in a translation-invariant way (e.g., choose with the uniform distribution or choose the least point in the lexicographic order). Let assign the value to the selected points and zero to the other points of . As in Example I.3.16, one can show that is an equivariant covering. Also, each point is covered at most times. So is -bounded (Definition I.3.12).
Theorem 2.16 (Minkowski Dimension in the Euclidean Case).
Let be a point-stationary point process in and assume the metric in is equivalent to the Euclidean metric. Then, for all equivariant weight functions such that a.s., one has
where is a uniformly at random point in independent of and .
By Theorem I.3.41, one may assume the metric on is the metric without loss of generality. Given any , consider the equivariant covering described above, but when choosing a random element of , choose point with probability (conditioned on ). One gets
As mentioned above, is equivariant and uniformly bounded (for all ). So Lemma I.3.13 implies both equalities in the claim. The inequalities are implied by the facts that and
which is implied by the Cauchy-Schwartz inequality. ∎
In many examples, the case where is used. An example where the decay rate of is strictly smaller than the growth rate of can be found in Example III.5.2 for suitable parameters. However, this example is not ergodic (see Remark I.3.28).
If is a point-stationary point process in and the metric on is equivalent to the Euclidean metric, then .
One may assume the metric on is the metric without loss of generality. Let and be a random point in chosen uniformly. For all discrete subsets and , let be the cube containing of the form (for ) and . Now, is an equivariant weight function. The construction readily implies that . Moreover, by , one has . Therefore, the unimodular Billingsley lemma (Theorem 2.8) implies that . ∎
If is a stationary point process in and is its Palm version, then
where is the intensity of .
Notice that if , the claim is implied by Theorem I.3.48. The general case is treated below.
For the first claim, by Proposition 2.17, it is enough to prove that . Let be a shifted square lattice independent of (i.e., , where is chosen uniformly, independently of ). Let . Since is a superposition of two independent stationary point processes, it is a stationary point process itself (see e.g., ). By letting , the Palm version of is obtained by the superposition of and an independent stationary lattice with probability (heads), and the superposition of and with probability (tails).
For all , there exists a disjoint -covering of (Example I.3.15). In both cases (heads or tails) above, one can consider this covering as a random subset of the shifted lattice. It is easy to see that it provides an equivariant -covering of (note that by enlarging the balls, all of is covered). Also, the probability of having a ball centered at the origin is . It follows that .
Note that has two natural equivariant subsets which, after conditioning to contain the origin, have the same distributions as and respectively. Therefore, one can use Theorem I.3.48 to deduce that . Therefore, Proposition 2.17 implies that .
Also, by using Theorem I.3.48 twice, one gets and . Therefore,
where the last equality is by Proposition I.3.38. So the claim is proved. ∎
The last claim of Proposition 2.18 suggests the following, which is verified when in the next proposition.
If is a point-stationary point process in which is not the Palm version of any stationary point process, then .
Conjecture 2.19 is true when .
Denote as such that and for each . Then, the sequence is stationary under shifting the indices (see Example I.2.18). The assumption that is not the Palm version of a stationary point process is equivalent to (see  or Proposition 6 of ). Indeed, if , then one could bias the probability measure by (Definition I.B.1) and then shift the whole process by , where is chosen uniformly and independently.
2.4 Connections to Birkhoff’s Pointwise Ergodic Theorem
This subsection discusses a corollary of the unimodular Billingsley lemma. The reader may skip it at first reading.
The following corollary of the unimodular Billingsley lemma is of independent interest. Note that the statement does not involve dimension.
Let be a unimodular discrete space. For any two equivariant weight functions and , if and is non-degenerate a.s., then
In particular, if and have well defined growth rates, then their growth rates are equal.
Note that the condition is necessary as shown in Remark 2.10.
Let be arbitrary and
It can be seen that is a measurable subset of . Assume . Denote by the random pointed discrete space obtained by conditioning on . Since does not depend on the root (i.e., if , then ), by a direct verification of the mass transport principle (I.2.2), one can show that is unimodular. So by using the unimodular Billingsley lemma (Theorem 2.8) twice, one gets
By the definition of , this contradicts the fact that a.s. So and the claim is proved. ∎
Theorem 2.21 is a generalization of a weaker form of Birkhoff’s pointwise ergodic theorem as explained in the following.
If is a point-stationary point process in , then the above argument still holds by using stationarity under shifting the origin to its next point (see Example I.2.18).
Note that amenability is not assumed in Theorem 2.21. But the claim is weaker since nothing can be said about .
The structure of this section is analogous to that of Section I.4. It provides further results on the examples introduced there. Some new examples are also presented.
3.1 General Unimodular Trees
3.1.1 Unimodular Two-Ended Trees
Here, another proof is given for the fact that the Minkowski and Hausdorff dimensions of any unimodular two-ended tree are equal to one (Theorem I.4.2) using the unimodular Billingsley lemma.
Let be a unimodular two-ended tree. For all two-ended trees and , let be 1 if belongs to the trunk of and 0 otherwise. It can be seen that is an equivariant process (Definition I.2.21). Let be the distance of to the trunk of . For larger than , one has . Therefore, the unimodular Billingsley lemma (Theorem 2.8) implies that . On the other hand, Proposition 2.14 implies that . Therefore, .
3.1.2 Unimodular Trees with Infinitely Many Ends
The following conjecture is of independent interest beyond its connections to the dimension. The authors believe it is new.
There is no unimodular random tree with polynomial growth and infinitely many ends a.s.
There is no equivariant metric (Definition I.3.40) on the -regular tree (for ) that has polynomial growth.
By the unimodular Billingsley lemma (Theorem 2.8), this conjecture is implied by Conjectures I.4.7 and I.4.8.
A stronger form of part (i) of the above conjecture is that there is no unimodular random tree with infinitely many ends a.s. such that .
The following example and proposition provide special cases where Conjecture I.4.8 is known to be true. Another example is provided in Subsection 3.2.1 below.
In the following, the ball of radius under the metric and centered at is denoted by .
By assigning i.i.d. random lengths to the edges on the -regular tree (as in Example I.2.24),
one gets an equivariant metric on , denoted by .
The claim is that for , one has ,
which implies that Conjecture I.4.8 holds in this case.
By Theorem I.3.41, one can replace the distribution of the lengths with another distribution which is stochastically larger. So one can assume the distribution of the lengths is non-lattice without loss of generality. Therefore, Theorem 21.1 of  for supercritical age-dependent branching processes implies that the limit of as exists a.s. and is positive for some . Therefore, Lemma 2.6 implies that and the claim is proved.
Recall from Subsection I.4.1.4 that an equivariant metric is said to be generated by equivariant edge lengths (or a geodesic metric) if for every path , one has .
Let be an equivariant metric on the 3-regular tree which is generated by equivariant edge lengths. If the random variable has finite moment of order , then . In particular, if it has finite moments of any order, then .
It is enough to assume for all since increasing the edge lengths does not increase the dimension (Theorem I.3.41). Consider the following equivariant weight function on :
where is a constant such that
It is easy to see that such a exists. Now, Lemma 3.4 below, which is a deterministic result, implies that a.s. for every . Also, the assumption on implies that has finite mean. So Lemma 2.6 implies that and the claim is proved. ∎
The following lemma is used in the last proof.
For , let be the infimum value of for all trees with the stated conditions. So one should prove . The claim is true for . Also, if , one has and the claim is trivial. The proof uses induction on . Assume that and for all , one has . For , let be the connected component containing when the edge is removed. It can be seen that