Unimodular Random Trees

TL;DR

This paper explores the properties of unimodular random trees and invariant forests, establishing their equivalence with certain percolation models, proving soficity, analyzing boundary behavior in hyperbolic geometry, and linking degree distribution integrability to convergence tightness.

Contribution

It demonstrates that bounded degree URTs are equivalent to invariant percolation components, provides a new proof of their soficity, and connects degree distribution properties to convergence behavior.

Findings

URTs of bounded degree correspond to invariant percolation components.

URTs are proven to be sofic via a new approach.

Ends of invariant forests in hyperbolic planes converge to boundary points.

Abstract

We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We also prove that uniform integrability of the degree distribution of a family of finite graphs implies tightness of that family for local convergence, also known as random weak convergence.