The martin boundary of a free product of abelian groups

Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green function of the corresponding random walk. It is known from the work of W. Woess that when a finitely supported random walk on a free product of abelian groups is adapted to the free product structure, the Martin boundary coincides with the geometric boundary. The main goal of this paper is to deal with non-adapted finitely supported random walks, for which there is no explicit formula for the Green function. Nevertheless, we show that the Martin boundary still coincides with the geometric boundary. We also prove that the Martin boundary is minimal

Entropy and drift for word metric on relatively hyperbolic groups

We are interested in the Guivarc'h inequality for admissible random walks on finitely generated relatively hyperbolic groups, endowed with a word metric. We show that for random walks with finite super-exponential moment, if this inequality is an equality, then the Green distance is roughly similar to the word distance, generalizing results of Blach{\`e}re, Ha{\"i}ssinsky and Mathieu for hyperbolic groups [4]. Our main application is for relatively hyperbolic groups with respect to virtually abelian subgroups of rank at least 2. We show that for such groups, the Guivarc'h inequality with respect to a word distance and a finitely supported random walk is always strict

The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups

Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups. In particular, in the case of nonuniform lattices in the real hyperbolic space H n , we show that the Martin boundary coincides with the CAT (0) boundary of the truncated space, and thus when n = 3, is homeomorphic to the Sierpinski carpet

Stability phenomena for Martin boundaries of relatively hyperbolic groups

Let Γ be a relatively hyperbolic group and let µ be an admissible symmetric finitely supported probability measure on Γ. We extend Floyd-Ancona type inequalities up to the spectral radius of µ. We then show that when the parabolic subgroups are virtually abelian, the Martin boundary of the induced random walk on Γ is stable in the sense of Picardello and Woess. We also define a notion of spectral degenerescence along parabolic subgroups and give a criterion for strong stability of the Martin boundary in terms of spectral degenerescence. We prove that this criterion is always satisfied in small rank. so that in particular, the Martin boundary of an admissible symmetric finitely supported probability measure on a geometrically finite Kleinian group of dimension at most 5 is always strongly stable

Stability phenomena for Martin boundaries of relatively hyperbolic groups

Let Γ be a relatively hyperbolic group and let µ be an admissible symmetric finitely supported probability measure on Γ. We extend Floyd-Ancona type inequalities up to the spectral radius of µ. We then show that when the parabolic subgroups are virtually abelian, the Martin boundary of the induced random walk on Γ is stable in the sense of Picardello and Woess. We also define a notion of spectral degenerescence along parabolic subgroups and give a criterion for strong stability of the Martin boundary in terms of spectral degenerescence. We prove that this criterion is always satisfied in small rank. so that in particular, the Martin boundary of an admissible symmetric finitely supported probability measure on a geometrically finite Kleinian group of dimension at most 5 is always strongly stable

An embedding of the Morse boundary in the Martin boundary

We construct a one-to-one continuous map from the Morse boundary of a hierarchically hyperbolic group to its Martin boundary. This construction is based on deviation inequalities generalizing Ancona's work on hyperbolic groups. This provides a possibly new metrizable topology on the Morse boundary of such groups. We also prove that the Morse boundary has measure 0 with respect to the harmonic measure unless the group is hyperbolic

Exotic local limit theorems at the phase transition in free products

We construct random walks on free products of the form Z 3 * Z d , with d = 5 or 6 which are divergent and not spectrally positive recurrent. We then derive a local limit theorem for these random walks, proving that $\mu$ * n (e) $\sim$ CR --n n --5/3 if d = 5 and $\mu$ * n (e) $\sim$ CR --n n --3/2 log(n) --1/2 if d = 6, where $\mu$ * n is the nth convolution power of $\mu$ and R is the inverse of the spectral radius of $\mu$. This disproves a result of Candellero and Gilch [7] and a result of the authors of this paper that was stated in a rst version of [11]. This also shows that the classication of local limit theorems on free products of the form Z d 1 * Z d 2 or more generally on relatively hyperbolic groups with respect to virtually abelian subgroups is incomplete

The growth of the Green function for random walks and Poincar{\'e} series

Given a probability measure $\mu$ on a finitely generated group $\Gamma$, the Green function $G(x,y|r)$ encodes many properties of the random walk associated with $\mu$. Finding asymptotics of $G(x,y|r)$ as $y$ goes to infinity is a common thread in probability theory and is usually referred as renewal theory in literature. Endowing $\Gamma$ with a word distance, we denote by $H_r(n)$ the sum of the Green function $G(e,x|r)$ along the sphere of radius $n$. This quantity appears naturally when studying asymptotic properties of branching random walks driven by $\mu$ on $\Gamma$ and the behavior of $H_r(n)$ as $n$ goes to infinity is intimately related to renewal theory. Our motivation in this paper is to construct various examples of particular behaviors for $H_r(n)$. First, our main result exhibits a class of relatively hyperbolic groups with convergent Poincar{\'e} series generated by $H_r(n)$, which answers some questions raised in a previous paper of the authors. Along the way, we investigate the behavior of $H_r(n)$ for several classes of finitely generated groups, including abelian groups, certain nilpotent groups, lamplighter groups, and Cartesian products of free groups