Co-spectral radius for countable equivalence relations

TL;DR

This paper introduces the co-spectral radius for inclusions of discrete probability measure-preserving equivalence relations, linking it to random walk growth and percolation theory, and develops new methods for analyzing unimodular random graphs.

Contribution

It defines the co-spectral radius for equivalence relation inclusions and extends the 2-3 method to prove walk growth existence on unimodular random graphs.

Findings

Co-spectral radius analogous to spectral radius in group theory.

Walk growth exists for unimodular random rooted graphs of bounded degree.

New critical exponents for percolation using co-spectral radius.

Abstract

We define the co-spectral radius of inclusions $\mathcal{S}\leq \mathcal{R}$ of discrete, probability measure-preserving equivalence relations, as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. For the proof, we develop a more general version of the 2-3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.