Canal-U

On the notion of dimension of unimodular discrete spaces (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

In this talk we will define notions of dimension for unimodular random graphs and point-stationary point processes. These notions are in spirit similar to the Minkowski dimension and the Hausdorff dimension. The key point in the definitions is the use of the mass transport principle which is used indispensably and distinguishes this view point from the previous notions which are defined in the literature. The connections of these definitions to volume growth and other notions of dimension are also discussed, which provide a toolset for calculating the dimension. Discrete analogues of several theorems regarding the dimension of continuum spaces are presented; e.g., the mass distribution principle, Billingsley’s lemma, Frostman’s lemma, and the max-flow min-cut theorem. In addition, the notion of unimodular discrete spaces is introduced which is a common generalization of unimodular random graphs and point-stationary point processes. The dimension of several examples of such spaces will be studied. Different methods for finding upper bounds and lower bounds on the dimension will also be presented and illustrated through these examples.