Co-spectral radius for countable equivalence relations

2022/05/13 by Abert, Miklós, Fraczyk, Mikolaj, Hayes, Ben · 1 citation
#05C81 #20F65 #37A30 #60K35 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2205.06692

Abstract

We define the co-spectral radius of inclusions S≤ 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≤ 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.

Cited by

• The Uniform Random Walk on graphs, loop processes and graphings

Related

• Coamenability and cospectral radius for orbit equivalence relations
• High Order Random Walks: Beyond Spectral Gap
• Walks and the spectral radius of graphs
• Co-spectral radius of intersections
• Tree walks and the spectrum of random graphs