We consider the harmonic measure on the Gromov boundary of a nonamenable hyperbolic group defined by a finite range random walk on the group, and study the corresponding orbit equivalence relation on the boundary. It is known to be always amenable and of type III. We determine its ratio set by showing that it is generated by certain values of the Martin kernel. In particular, we show that the equivalence relation is never of type III_0.
We consider random walks on a non-elementary hyperbolic group endowed with a word distance. To a probability measure on the group are associated two numerical quantities, the rate of escape and the entropy. On the set of admissible probability measures whose support is contained in a given finite set, we show that both quantities depend in an analytic way on the probability measure. Our spectral techniques also give a new proof of the central limit theorem, and imply that the corresponding variance is analytic.
We consider nondegenerate, finitely supported random walks on a finitely generated Gromov hyperbolic group. We show that the entropy and the escape rate are Lipschitz functions of the probability if the support remains constant.
Let $Γ$ be a countable group acting on a geodesic Gromov-hyperbolic metric space $X$ and $μ$ a probability measure on $Γ$ whose support generates a non-elementary subsemigroup. Under the assumption that $μ$ has a finite exponential moment, we establish large deviations results for the distance and the translation length of a random walk with driving measure $μ$. From our results, we deduce a special case of a conjecture regarding large deviations of spectral radii of random matrix products.
We show exact dimensionality of harmonic measures associated with random walks on groups acting on a hyperbolic space under finite first moment condition, and establish the dimension formula by the entropy over the drift. We also treat the case when a group acts on a non-proper hyperbolic space acylindrically. Applications of this formula include continuity of the Hausdorff dimension with respect to driving measures and Brownian motions on regular coverings of a finite volume Riemannian manifold.
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{è}re, Ha{ï}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 t…
We study random walks on relatively hyperbolic groups whose law is convergent, in the sense that the derivative of its Green function is finite at the spectral radius.When parabolic subgroups are virtually abelian, we prove that for such a random walk satisfies a local limit theorem of the form $p_n(e, e)\sim CR^{-n}n^{-d/2}$, where $p_n(e, e)$ is the probability of returning to the origin at time $n$, $R$ is the inverse of the spectral radius of the random walk and $d$ is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate.This concludes the classific…
We prove that a probability measure on a compact non-singular lamination by hyperbolic Riemann surfaces is harmonic if and only if it is the projection of a measure on the unit tangent bundle such that it is invariant under both the geodesic and the horocycle flows.
Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk. If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with …
This paper deals with random walks on isometry groups of Gromov hyperbolic spaces, and more precisely with the dimension of the harmonic measure $ν$ associated with such a random walk. We first establish a link of the form $\dim ν\leq h/l$ between the dimension of the harmonic measure, the asymptotic entropy $h$ of the random walk and its rate of escape $l$. Then we use this inequality to show that the dimension of this measure can be made arbitrarily small and deduce a result on the type of the harmonic measure.
This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive-recurrence for random walks on relatively hy-perbolic groups. We then use our estimates for the Green function to prove that $p_n \asymp R^n n^{-3/2}$ for spectrally positive-recurrent random walks, w…
It is known that every infinite index quasi-convex subgroup $H$ of a non-elementary hyperbolic group $G$ is a free factor in a larger quasi-convex subgroup of $G$. We give a probabilistic generalization of this result. That is, we show that when $R$ is a subgroup generated by independent random walks in $G$, then $\langle H, R\rangle\cong H\ast R$ with probability going to one as the lengths of the random walks go to infinity and this subgroup is quasi-convex in $G$. Moreover, our results hold for a large class of groups acting on hyperbolic metric spaces and subgroups with quasi-convex orbits…
In this paper, we extend a result of Kesten and Spitzer (1979). Let us consider a stationary sequence $(ξ\_k:=f(T^k(.)))\_k$ given by an invertible probability dynamical system and some centered function $f$. Let $(S\_n)\_n$ be a simple symmetric random walk on $Z$ independent of $(ξ\_k)\_k$. We give examples of partially hyperbolic dynamical systems and of functions $f$ such that $n^{-3/4}(ξ(S\_1)+...+ξ(S\_k))$ converges in distribution as $n$ goes to infinity.
In this work we prove the continuity and existence of large deviations for the drift of random walks on groups acting by isometries on Gromov Hyperbolic Spaces. Through the process we refine the multiplicative ergodic theorem of Karlsson and Gouëzel for such spaces. The works goes beyond what is known in the literature by allowing spaces that are not necessarily proper.
Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.
For a group hyperbolic relative to virtually nilpotent subgroups, on a cusped graph associated to the group, we construct a random walk whose Martin boundary is the Bowditch boundary of the group. Moreover, the harmonic measure is a conformal density corresponding to a hyperbolic Green metric and is exact dimensional on the Bowditch boundary. The latter equipped with a visual distance induced by the Green metric is an Ahlfors-regular metric measure space. The dimension is given in terms of the drift, a Green drift and the asymptotic entropy. The Patterson-Sullivan density for the action on the…
Completing a strategy of Gouëzel and Lalley, we prove a local limit theorem for the random walk generated by any symmetric finitely supported probability measure on a non-elementary Gromov-hyperbolic group: denoting by $R$ the inverse of the spectral radius of the random walk, the probability to return to the identity at time $n$ behaves like $C R^{-n}n^{-3/2}$. An important step in the proof is to extend Ancona's results on the Martin boundary up to the spectral radius: we show that the Martin boundary for $R$-harmonic functions coincides with the geometric boundary of the group. In an append…
The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures, so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model and establish an analogue of the Poisson formula for (random) bounded harmonic functions. Under natural conditions on transition probabilities we identify t…
We show that the equidistribution theorem of C. Bonatti and X. Gómez-Mont for a special kind of foliations by hyperbolic surfaces does not hold in general, and seek for a weaker form valid for general foliations by hyperbolic surfaces.
We consider harmonic measures that arise from random walks on the mapping class group determined by probability distributions that have finite first moment with respect to the Teichmuller metric, and whose supports generate non-elementary subgroups. We prove that Teichmuller space with the Teichmuller metric is statistically hyperbolic for such a harmonic measure.
We prove that the hitting measure is not equivalent to the Lebesgue measure for a large class of nearest-neighbour random walks on hyperbolic reflection groups and Fuchsian groups.
We prove that for suitable random walks on isometry groups of $CAT(-1)$ spaces, typical sample paths eventually land on loxodromic elements which equidistribute with respect to a flow invariant measure on the unit tangent bundle of the quotient space.
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 the first paper to show that $p_n(e, e) \sim CR^{-n} n^{-3/2}$ , where $R$ is the spectral radius of the random walk. This generalizes results of W. Woess for free products and r…
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.
We consider the harmonic measure on a disconnected polynomial Julia set in terms of Brownian motion. We show that the harmonic measure of any connected component of such a Julia set is zero. Associated to the polynomial is a combinatorial model, the tree with dynamics. We define a measure on the tree, which is a combinatorial version on harmonic measure. We show that this measure is isomorphic to the harmonic measure on the Julia set. The measure induces a random walk on the tree, which is isomorphic to Brownian motion in the plane.
Given integers $d \geq 2, n \geq 1$, we consider affine random walks on torii $(\mathbb{Z} / n \mathbb{Z})^{d}$ defined as $X_{t+1} = A X_{t} + B_{t} \mod n$, where $A \in \mathrm{GL}_{d}(\mathbb{Z})$ is an invertible matrix with integer entries and $(B_{t})_{t \geq 0}$ is a sequence of iid random increments on $\mathbb{Z}^{d}$. We show that when $A$ has no eigenvalues of modulus $1$, this random walk mixes in $O(\log n \log \log n)$ steps as $n \rightarrow \infty$, and mixes actually in $O(\log n)$ steps only for almost all $n$. These results generalize those on the so-called Chung-Diaconis-G…
Let $S=Γ\backslash \mathbb{H}$ be a hyperbolic surface of finite topological type, such that the Fuchsian group $Γ\le \operatorname{PSL}_2(\mathbb{R})$ is non-elementary, and consider any generating set $\mathfrak S$ of $Γ$. When sampling by an $n$-step random walk in $π_1(S) \cong Γ$ with each step given by an element in $\mathfrak S$, the subset of this sampled set comprised of hyperbolic elements approaches full measure as $n\to \infty$, and for this subset, the distribution of geometric lengths obeys a Law of Large Numbers, Central Limit Theorem, Large Deviations Principle, and Local Limit…
Let $Γ$ be a nonelementary hyperbolic group with a word metric $d$ and $\partialΓ$ its hyperbolic boundary equipped with a visual metric $d_a$ for some parameter $a>1$. Fix a superexponential symmetric probability $μ$ on $Γ$ whose support generates $Γ$ as a semigroup, and denote by $ρ$ the spectral radius of the random walk $Y$ on $Γ$ with step distribution $μ$. Let $ν$ be a probability on $\{1,\, 2, \, 3, \, \ldots\}$ with mean $λ=\sum\limits_{k=1}^\infty kν(k)<\infty$. Let $\mathrm{BRW}(Γ, \, ν, \, μ)$ be the branching random walk on $Γ$ with offspring distribution $ν$ and base motion $Y$ an…
We generalize the notion of cusp excursion of geodesic rays by introducing for any $k \geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic $N$-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for $k = N-1$, the $k^{th}$ excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space $\mathbb{H}^N$ for any $N \geq 2…
