Processes on Unimodular Random Networks

Abstract

We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications to stochastic comparison of continuous-time random walk. Contents2000 Mathematics Subject Classification. Primary 60C05. Secondary 60K99, 05C80.

Ultraproducts of measure preserving actions and graph combinatorics

“…Let d be the degree of Cay( , S). In Aldous and Lyons [AL,10.5], the question of existence of invariant, random k-colorings is discussed and it was mentioned that Schramm (unpublished, 1997) had shown that for any , S there is an invariant, random (d + 1)-coloring (this also follows from the more general Kechris-Solecki-Todorcevic approach [KST,4.8]). They also pointed out that Brooks' theorem implies that there is an invariant, random d-coloring when is a sofic group (for the definition of sofic group, see e.g.…”

Hyperlinear and Sofic Groups: A Brief Guide

“…Sofic groups have been linked to theory of stochastic processes in infinite networks [3] and to cellular automation [15], and have been shown to admit a classification of their Bernoulli actions [12].…”

Coarse Geometry and Randomness

“…It is also possible to go from a stationary and reversible random graph towards a unimodular random one by biasing by deg(ρ) −1 , see [AL07,BC12].…”

“…This is interesting, because of course it does not have translation invariance of any form. But unimodularity can replace that in many arguments, see [AL07]. How does IIC looks like on a 2 − 3-tree?…”