Local limit theorems in relatively hyperbolic groups II: the non-spectrally degenerate case

Abstract

This is the second of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this second paper, we restrict our attention to non-spectrally degenerate random walks and we prove precise asymptotics of the probability $p_n(e,e)$ of going back to the origin at time $n$ . We combine techniques adapted from thermodynamic formalism with the rough estimates of the Green function given by part I to show that $p_n(e,e)\sim CR^{-n}n^{-3/2}$ , where $R$ is the inverse of the spectral radius of the random walk. This both generalizes results of Woess for free products and results of Gouëzel for hyperbolic groups.

Ratio-limit boundaries of random walks on relatively hyperbolic groups

“…We will also need the following deviation inequality for random walks on relatively hyperbolic groups, which we call the strong relative Ancona inequality. These were proved by the second and third named authors [31,Theorem 3.14] and [27,Theorem 2.15]. For hyperbolic groups, these were first proved for by Izumi-Neshveyev-Okayasu [46] for r = 1 and by Gouezël [38] uniformly for r ≤ R.…”

“…Spectral non-degeneracy is of particular importance in the study of random walks on relatively hyperbolic groups. For instance, it is a determining property characterizing the homeomorphism type of the R-Martin boundary as shown in [31], as well as a key property for establishing local limit theorems [27]. Regardless of whether or not µ is spectrally non-degenerate, it follows from equation (4.1) and [31, Lemma 6.2, Proposition 6.3], respectively, that…”

“…To arrive at a formula for the ratio-limit kernels involving in terms of the expressions I (s) (x, y|r), we will need to relate the asymptotic behavior for higher derivatives of Green functions with that of P n (x, y). This was established by the second author and collaborators in a sequence of papers [26,27,29].…”

“…It is not known whether these limits exist in general, and when they do we say that the random walk determined by µ has the strong ratio-limit property (SRLP). It is known that SRLP holds for random walks on nilpotent groups [52], symmetric random walks on amenable groups [5], symmetric random walks on non-elementary hyperbolic groups [38], and certain large classes of random walks on relatively hyperbolic groups [27,29].…”

“…Despite these aforementioned limitations, the strategy employed by Woess in [68] can still be adapted for many random walks on relatively hyperbolic groups, including spectrally non-degenerate random walks. In Section 6 we employ results of the second author from [26,27] to show that for a spectrally degenerate random walk on a relatively hyperbolic group Γ, for a point ξ ∈ ∂ M,R Γ whose image in Bowditch boundary is conical, we have that H(x,y) K R (x,y) → 1 as y → ξ. This allows us show in Corollary 6.2 that closure of minimal points ∂ m M,R Γ embed inside ratio-limit boundary ∂ ρ Γ via a bi-Lipschitz Γ-equivariant map.…”