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Unimodular random trees

Published online by Cambridge University Press:  20 August 2013

ITAI BENJAMINI ,
RUSSELL LYONS  and
ODED SCHRAMM
ITAI BENJAMINI
Affiliation:
Mathematics Department, The Weizmann Institute of Science, Rehovot 76100, Israel email Itai.Benjamini@weizmann.ac.il
RUSSELL LYONS
Affiliation:
Department of Mathematics, 831 E 3rd St, Indiana University, Bloomington, IN 47405-7106, USA email rdlyons@indiana.edu
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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.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 2 , April 2015 , pp. 359 - 373
Copyright
© Cambridge University Press, 2013 

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