Graphings and Unimodularity

TL;DR

This paper extends the concept of graph laws to infinite graphings with probabilistic vertices, generating new unimodular measures and proving that weak limits preserve unimodularity, thus broadening the theoretical framework of graph limits.

Contribution

It introduces a generalized framework for graph laws to infinite graphings and provides a detailed proof that weak limits maintain unimodularity.

Findings

New unimodular measures from infinite graphings

Weak limits preserve unimodularity in graph sequences

Extended the theoretical understanding of graph limits

Abstract

We extend the concept of the law of a finite graph to graphings, which are, in general, infinite graphs whose vertices are equipped with the structure of a probability space. By doing this, we obtain a vast array of new unimodular measures. Furthermore, we work out in full detail a proof of a known result, which states that weak limits preserve unimodularity.