Unimodular Hausdorff and Minkowski Dimensions

TL;DR

This paper introduces unimodular Minkowski and Hausdorff dimensions for discrete spaces, providing a unified framework that generalizes classical notions and connects to various probabilistic and geometric properties.

Contribution

It defines new unimodular dimensions for discrete spaces, develops analytical tools like analogues of Billingsley's and Frostman's lemmas, and applies these to diverse stochastic models.

Findings

Unimodular dimensions relate to volume growth and scaling limits.

New bounds for dimensions using the developed lemmas.

Applications to point processes, random graphs, and self-similar spaces.

Abstract

This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced in this work, which provide a common generalization to stationary point processes under their Palm version and unimodular random rooted graphs. The use of unimodularity in the definitions of dimension is novel. Also, a toolbox of results is presented for the analysis of these dimensions. In particular, analogues of Billingsley's lemma and Frostman's lemma are presented. These last lemmas are instrumental in deriving upper bounds on dimensions, whereas lower bounds are obtained from specific coverings. The notions of unimodular Hausdorff size, which is a discrete analogue of the Hausdorff measure, and unimodular dimension function are also introduced. This toolbox allows one to connect the unimodular dimensions to other notions such as volume growth rate, discrete dimension and scaling limits. It is also used to analyze the dimensions of a set of examples pertaining to point processes, branching processes, random graphs, random walks, and self-similar discrete random spaces. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces.