Ratio limits and Martin boundary

Abstract

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

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

“…In this paper, we introduce a new Cuntz-type C*-algebra O(G, µ) for a random walk P on a group G induced by a finitely supported measure µ, which is a quotient of the Toeplitz algebra T (G, µ) of the stochastic matrix P . The computation of O(G, µ) in this paper gave rise to new notions of ratio-limit space and boundary for random walks, prompting the study in the companion paper by Woess [58]. When the random walk is finite, our Cuntz C*-algebras coincide with the ones computed in [20,Theorem 2.1], but new subtleties emerge for random walks on infinite groups.…”

“…We answer the above question in the negative, showing that Viselter's Cuntz-Pimsner C*-algebra has a proper quotient which is the unique G×T equivariant quotient of T (G, µ). In fact, for symmetric aperiodic random walks on nonelementary hyperbolic groups, whose ratio-limit boundary is computed in the companion paper [58], we get a unique G × T-equivariant quotient of T (G, µ) which is a proper quotient of both Viselter's C*-algebra and O(G, µ).…”

“…Local limit theorems have been established for certain random walks on free products [54,8], random walks on free groups and trees [37], symmetric random walks on co-compact Fuchsian groups [29] and symmetric random walks on non-elementary hyperbolic groups [28]. For more on the history of local limit theorems we refer the reader to [56, Chapter III], as well as the companion paper by Woess [58]. In general, it is unknown whether or not aperiodic random walks automatically satisfy SRLP.…”

“…This result prompted the computation of the ratio-limit boundary for several classes of examples in the companion paper by Woess [58]. This includes isotropic random walks on trees [58,Theorem 3.3], random walks on free groups [58,Theorem 3.12], and symmetric random walks on non-elementary hyperbolic groups [58,Theorem 4.5].…”

“…In Section 2 we provide some of the necessary preliminaries on stochastic matrices and random walks. In Section 3 we introduce the ratio-limit space and boundary of a random walk with SRLP arising in this work, and provide some examples by appealing to the companion paper by Woess [58]. In Section 4 we define Toeplitz and Cuntz algebras for random walks, and compute the latter under the assumption of SRLP.…”