Unimodular Random Trees

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.

19 pages, 4 figures

References in corpus (2)

• The Canopy Graph and Level Statistics for Random Operators on Trees
• Exponential tail bounds for loop-erased random walk in two dimensions

Cited by in corpus (10)

• Large deviations of empirical neighborhood distribution in sparse random graphs
• Eternal Family Trees and Dynamics on Unimodular Random Graphs
• Unimodular measures on the space of all Riemannian manifolds
• Random intersection graphs with communities
• Interacting growth processes and invariant percolation
• Steady state clusters and the Rath-Toth mean field forest fire model
• Invariant embeddings of unimodular random planar graphs
• Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit
• Age evolution in the mean field forest fire model via multitype branching processes
• How Well Do Local Algorithms Solve Semidefinite Programs?