On Weak Limits and Unimodular Measures

TL;DR

This thesis explores probability measures on countable rooted graphs, focusing on unimodular measures, their properties, and limits, providing new criteria, uniqueness results, and counterexamples to conjectures.

Contribution

It introduces new criteria for unimodularity, proves uniqueness of unimodular measures on connected graphs, and constructs counterexamples to conjectures about weak limits.

Findings

Connected graphs sustain at most one unimodular measure

Unimodular measures on disconnected graphs are convex combinations

Counterexamples to conjectures on weak limits of finite graph laws

Abstract

In this thesis, the main objects of study are probability measures on the isomorphism classes of countable, connected rooted graphs. An important class of such measures is formed by unimodular measures, which satisfy a certain equation, sometimes referred to as the intrinsic mass transport principle. The so-called law of a finite graph is an example of a unimodular measure. We say that a measure is sustained by a countable graph if the set of rooted connected components of the graph has full measure. We demonstrate several new results involving sustained unimodular measures, and provide thorough arguments for known ones. In particular, we give a criterion for unimodularity on connected graphs, deduce that connected graphs sustain at most one unimodular measure, and prove that unimodular measures sustained by disconnected graphs are convex combinations. Furthermore, we discuss weak limits of laws of finite graphs, and construct counterexamples to seemingly reasonable conjectures.