Ratio limits and Martin boundary

TL;DR

This paper explores the relationship between the ratio limit theorem, the spectral radius, and the Martin boundary in irreducible Markov chains, with a focus on random walks on non-amenable groups like free and hyperbolic groups.

Contribution

It establishes connections between the $ ho$-Martin boundary and the boundary from the $ ho$-harmonic kernel in the context of non-amenable groups.

Findings

Identifies the relationship between the $ ho$-Martin boundary and harmonic kernels.

Provides insights into random walks on free and hyperbolic groups.

Enhances understanding of boundary behavior in non-amenable group contexts.

Abstract

Consider an irreducible Markov chain which satisfies a ratio limit theorem, and let $\rho$ be the spectral radius of the chain. We investigate the relation of the the $\rho\,$-Martin boundary with the boundary induced by the $\rho\,$-harmonic kernel which appears in the ratio limit. Special emphasis is on random walks on non-amenable groups, specifically, free groups and hyperbolic groups.