Space-time boundaries for random walks and their application to operator algebras

TL;DR

This paper explores the space-time Martin boundary of random walks on groups, relating it to classical compactifications, and applies these findings to operator algebras, revealing structural insights and boundaries.

Contribution

It introduces the $0$-Martin boundary, relates it to classical compactifications, and identifies the minimal space-time Martin boundary with a union of $ ext{Martin}$ boundaries, connecting probabilistic and operator algebraic structures.

Findings

The reduced ratio-limit compactification embeds into the space-time Martin boundary.

The $0$-Martin boundary governs $ ext{infty}$-harmonic functions and arises as limits of $ ext{Martin}$ kernels.

The noncommutative Shilov boundary coincides with the Toeplitz $C^*$-algebra.

Abstract

We investigate the Martin boundary of the space-time Markov chain associated to a finitely supported random walk $(\Gamma, \mu)$ with spectral radius $\rho$ and relate it to several classical compactifications of $\Gamma$. Assuming the strong ratio-limit property, we prove that the reduced ratio-limit compactification embeds naturally into the space-time Martin boundary. We introduce the $0$-Martin boundary, which governs the behaviour of $\infty$-harmonic functions, and show that the $0$-Martin kernels arise as rescaled limits of $\lambda$-Martin kernels as $\lambda\rightarrow 0$. For symmetric random walks on hyperbolic groups, the $0$-Martin boundary naturally covers the Gromov boundary, while the cover need not be injective in general. Our main structural theorem identifies the minimal space-time Martin boundary with the disjoint union of minimal $\lambda$-Martin boundaries over $\lambda\in [0, \rho^{-1}]$ with its natural pointwise topology. As an application, we show that the noncommutative Shilov boundary of the tensor algebra of the random walk $(\Gamma, \mu)$ coincides with its Toeplitz $C^*$-algebra.