Toeplitz quotient C*-algebras and ratio limits for random walks

TL;DR

This paper introduces a new C*-algebra framework for analyzing random walks with convergent transition probability ratios, revealing new ratio limit spaces and boundaries, especially for symmetric walks on hyperbolic groups.

Contribution

It develops a novel Cuntz-type quotient C*-algebra for specific random walks and identifies a unique minimal symmetry-equivariant quotient for symmetric hyperbolic group walks.

Findings

New quotient C*-algebras for ratio-convergent random walks

Definition of ratio limit space and boundary for these walks

Identification of a minimal symmetry-equivariant quotient for hyperbolic groups

Abstract

We study quotients of the Toeplitz C*-algebra of a random walk, similar to those studied by the author and Markiewicz for finite stochastic matrices. We introduce a new Cuntz-type quotient C*-algebra for random walks that have convergent ratios of transition probabilities. These C*-algebras give rise to new notions of ratio limit space and boundary for such random walks, which are computed by appealing to a companion paper by Woess. Our combined results are leveraged to identify a unique smallest symmetry-equivariant quotient C*-algebra for any symmetric random walk on a hyperbolic group, shedding light on a question of Viselter on C*-algebras of subproduct systems.