Regularity of the entropy for random walks on hyperbolic groups

Ledrappier, François

doi:10.1214/12-aop748

Cited by 17 publications

(17 citation statements)

References 19 publications

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“…This implies the desired estimate on G R (x, z; B(y, n) c ) by Cauchy-Schwarz, as follows. To write it formally, it is more convenient to express things in terms of operators, as in [Led11]. Define an operator L i :…”

Section: Ancona Inequalitiesmentioning

confidence: 99%

Local limit theorem for symmetric random walks in Gromov-hyperbolic groups

Gouëzel¹

2014

J. Amer. Math. Soc.

137

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Completing a strategy of Gouëzel and Lalley [GL11], 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 CR −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 appendix, we explain how the symmetry assumption of the measure can be dispensed with for surface groups.

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Section: Ancona Inequalitiesmentioning

confidence: 99%

Local limit theorem for symmetric random walks in Gromov-hyperbolic groups

Gouëzel¹

2014

J. Amer. Math. Soc.

137

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“…Analyticity of the rate of escape and of the asymptotic entropy on free groups was proven by Ledrappier in [25]. More recently, Ledrappier [26] proves Lipschitz continuity for the rate of escape and asymptotic entropy for random walks on Gromov hyperbolic groups.…”

Section: Analyticity Of the Rate Of Escape And Asymptotic Variancementioning

confidence: 95%

Renewal theory for random walks on surface groups

Haı̈ssinsky¹,

Mathieu²,

Müller³

2016

Ergod. Th. Dynam. Sys.

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Abstract. We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enter a particular type of cone and never leave it again. As a consequence, the trajectory of the random walk can be expressed as an aligned union of independent and identically distributed trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.

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“…This fact applies, in particular, to the case of bounded range random walks on virtually free groups, which goes beyond the scope of previous results related to the question of analyticity. At this point let us summarize several papers concerning continuity and analyticity of the drift and entropy that have been published recently: e.g., see Ledrappier [15], [16], Haïssinsky, Mathieu and Müller [9], Gilch [7]. The recent survey article of Gilch and Ledrappier [5] collects several results about analyticity of drift and entropy of random walks on groups.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic entropy of random walks on regular languages over a finite alphabet

Gilch¹

2016

Electron. J. Probab.

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We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant support, which describe the random walk. This setting applies, in particular, to random walks on virtually free groups.the concatenated word. A random walk on a regular language is a Markov chain (X n ) n∈N 0 on the set A * = n≥1 A n ∪ {o}, whose transition probabilities obey the following rules:(i) Only the last two letters of the current word may be modified.(ii) Only one letter may be adjoined or deleted at one instant of time. (iii) Adjunction and deletion may only be done at the end of the current word. (iv) Probabilities of modification, adjunction or deletion depend only on the last two letters of the current word and on the substitute letters. Compare with Lalley [14]and Gilch [6]. In other words, at each step the last two letters of the current word may be replaced by a non-empty word of length of at most 3 and the transition probabilities depend only on the last two letters of the current word and the replacing word of length of at most 3. More formally, the transition probabilities of the Markov chain (X n ) n∈N 0 can be written as follows, where w ∈ A * , a 1 , a 2 , b 1 , b 2 , b 3 ∈ A:Not all of these probabilities need to be strictly positive. Initially, we set X 0 := o. If we start the random walk at w ∈ A * instead of o, we write P w [ · ] := P[ · | X 0 = w]. For w 1 , w 2 ∈ A * , the n-step transition probabilities are denoted by p (n) (w 1 , w 2 ) := P w 1 [X n = w 2 ]. The set of accessible words from o is given byWe will also think of the random walk (X n ) n∈N 0 as a nearest neighbour random walk on an undirected graph G, where the vertices are the elements of L and undirected edges are between two vertices if and only if one can walk from one word to the other one in a single step. For this purpose, we need the following assumption:We call this property weak symmetry.In particular, Assumption 2.1 yields irreducibility of the random walk on L. Moreover, this assumption will be necessary for the construction of a sequence of cones in the graph G which track the random walk's way to infinity. As the interested reader will see, weak symmetry can obviously be weakened in some way but for reason of better readability we keep this natural assumption; for a discussion on this assumption, we refer to Appendix A.2.Since the purpose of the paper is the investigation of the asymptotic behaviour of transient random walks, we obviously need that L is infinite in our setting. It is an easy exercise

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Regularity of the entropy for random walks on hyperbolic groups

Cited by 17 publications

References 19 publications

Local limit theorem for symmetric random walks in Gromov-hyperbolic groups

Local limit theorem for symmetric random walks in Gromov-hyperbolic groups

Renewal theory for random walks on surface groups

Asymptotic entropy of random walks on regular languages over a finite alphabet

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