Asymptotic Entropy of Random Walks on Free Products

Gilch, Lorenz A.

doi:10.1214/ejp.v16-841

Cited by 13 publications

(25 citation statements)

References 24 publications

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“…The question of almost sure convergence of − 1 n log π n (X n ) to some constant h, however, remains open. Similar results concerning existence and formulas for the entropy are proved in Gilch and Müller [8] for random walks on directed covers of graphs and in Gilch [7] for random walks on free products of graphs. Furthermore, we give formulas for the entropy which allow numerical computations and also exact calculations in some special cases.…”

Section: Introductionsupporting

confidence: 79%

“…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%

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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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Section: Introductionsupporting

confidence: 79%

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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“…The framework of our proofs follows ideas from Gilch [11], where the entropy for random walks on regular languages is investigated. Similar results, in different contexts, concerning existence of the entropy are proved in Gilch and Müller [15] for random walks on directed covers of graphs, and in Gilch [14] for random walks on free products of graphs. Moreover, a survey article on rate of escape and entropy of random walks is Gilch and Ledrappier [12] and, in particular, for random walks on hyperbolic groups see Ledrappier [18].…”

Section: Introductionsupporting

confidence: 76%

“…We follow the reasoning of [14] for the proof of existence of the entropy. First, we remark that due to irreducibility of the retracted walk and by [13,Proposition 4.6,Corollary 4.9] we have a unique radius of convergence R > 1 of G(u, v|z) for all u, v ∈ W .…”

Section: The Entropy Of the Retracted Walkmentioning

confidence: 99%

Asymptotic entropy of random walks on Fuchsian buildings and Kac–Moody groups

Gilch¹,

Müller²,

Parkinson³

2016

Math. Z.

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In this article we prove existence of the asymptotic entropy for isotropic random walks on regular Fuchsian buildings. Moreover, we give formulae for the asymptotic entropy, and prove that it is equal to the rate of escape of the random walk with respect to the Green distance. When the building arises from a Fuchsian Kac-Moody group our results imply results for random walks induced by bi-invariant measures on these groups, however our results are proven in the general setting without the assumption of any group acting on the building. The main idea is to consider the retraction of the isotropic random walk onto an apartment of the building, to prove existence of the asymptotic entropy for this retracted walk, and to 'lift' this in order to deduce the existence of the entropy for the random walk on the building.

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“…Moreover, it follows from the formula in [15] that the escape rate is Lipschitz continuous in on P(F ); see the remark after Formula (4) below. This holds more generally for free products; see [10] and [11] for the precise conditions. This holds more generally for free products; see [10] and [11] for the precise conditions.…”

Section: Introductionmentioning

confidence: 98%

Regularity of the entropy for random walks on hyperbolic groups

Ledrappier

2013

Ann. Probab.

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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.Comment: Published in at http://dx.doi.org/10.1214/12-AOP748 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Asymptotic Entropy of Random Walks on Free Products

Cited by 13 publications

References 24 publications

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

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

Asymptotic entropy of random walks on Fuchsian buildings and Kac–Moody groups

Regularity of the entropy for random walks on hyperbolic groups

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