On the Dimension of Unimodular Discrete Spaces

Abstract: This talk is focused on large scale properties of infinite graphs and discrete subsets of the Euclidean space. We present two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired by the classical Minkowski and Hausdorff dimensions. These dimensions are defined for unimodular discrete spaces, which are defined in this work as a class of random discrete metric spaces with a distinguished point called the origin. These spaces provide a common generalization to stationary point processes (under their Palm version) and unimodular random rooted graphs.

The concept of unimodularity, developed for graphs in the last two decades, can be interpreted as statistical homogeneity. Unimodular graphs appear in many contexts, such as graph limits, Cayley graphs, transitive graphs with a unimodular automorphism group, and stationary point processes.

If one wants to cover an infinite discrete space by large balls, infinitely many balls are needed anyway. The main novelty is the use of unimodularity in the definitions where it suggests replacing the infinite sums regarding the coverings by the expectation of certain random variables at the origin. Other related notions such as volume growth are also discussed, which provide tools to calculate the dimension.  Several examples will be discussed in relation with the theory of point processes, unimodular graphs and self-similarity. Different methods for finding upper bounds and lower bounds on the dimension will also be presented and illustrated through the examples.

This work is a joint work with Francois Baccelli and Mir-Omid Haji-Mirsadeghi.

Ali Khezeli