Unimodular random trees

By Itai Benjamini, Russell Lyons, Oded Schramm. Ergodic Theory and Dynamical Systems, 2012

68 citations recorded by the source corpus.

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 note 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f22d36e9f5db7d1da31b8bcd9d1f777d02e5b78b

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Hyperbolic and Parabolic Unimodular Random Maps (2016, methodology): “y.) One main ingredient toTheorem 2is the following, which was proven for Cayley graphs of free groups by [24]. Theorem 8.1 (Bowen [24]; Elek [32]; Elek and Lippner [33]; Benjamini, Lyons and Schramm [16]). Every unimodular random rooted tree is strongly soc. The following theorem is an adaptation o”
Large deviations of empirical neighborhood distribution in sparse random graphs (2013, methodology): “ction 2 we review some basic facts about local weak convergence in the context of multigraphs. We also establish a compactness criterion which parallels recent results of Benjamini, Lyons and Schramm [3]. In Section 3 we introduce the unimodular Galton Watson trees with given h-neighborhood distribu”