On Continuity of Range, Entropy and Drift for Random Walks on Groups

Erschler, Anna

doi:10.1007/978-3-0346-0244-0_3

Cited by 9 publications

(5 citation statements)

References 9 publications

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“…Erschler and Kaimanovich [9] asked if drift and entropy of random walks on groups depend continuously on the probability measure, which governs the random walk. Ledrappier [19] proves in his recent, simultaneous paper that drift and entropy of finite-range random walks on free groups vary analytically with the probability measure of constant support.…”

Section: Entropy Of Random Walks On Free Products Of Groupsmentioning

confidence: 99%

“…with Kaimanovich and Vershik [14] and Erschler [8]. Erschler and Kaimanovich [9] asked if drift and entropy of random walks on groups vary continuously on the probability measure, which governs the random walk. We prove real-analyticity of the entropy when varying the probabilty measure of constant support; compare also with the recent work of Ledrappier [19], who simultaneously proved this property for finite-range random walks on free groups.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Entropy of Random Walks on Free Products

Gilch

2011

Electron. J. Probab.

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Abstract. Suppose we are given the free product V of a finite family of finite or countable sets. We consider a transient random walk on the free product arising naturally from a convex combination of random walks on the free factors. We prove the existence of the asymptotic entropy and present three different, equivalent formulas, which are derived by three different techniques. In particular, we will show that the entropy is the rate of escape with respect to the Greenian metric. Moreover, we link asymptotic entropy with the rate of escape and volume growth resulting in two inequalities.

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Section: Entropy Of Random Walks On Free Products Of Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Asymptotic Entropy of Random Walks on Free Products

Gilch

2011

Electron. J. Probab.

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“…Only in a handful of examples, can we explicitly compute (µ; d). In [EK13], it is proved that, for nonelementary hyperbolic groups and under a first moment assumption, then the asymptotic entropy is continuous for the weak topology on measures -a fact that fails to be true in all groups, see [Ers11]. If we restrict ourselves to measures µ with fixed finite support (and still assume that G is non-elementary hyperbolic), F. Ledrappier proved in [Led13] that h and are Lipschitz continuous.…”

Section: Lipschitz Continuity and Differentiability Of The Rate Of Es...mentioning

confidence: 99%

“…The question of the regularity of the rate of escape or the entropy in terms of the driving measure was first addressed in [Ers11]. Gouëzel recently proved in [Gou15] that, on a hyperbolic group G, the entropy and rate of escape are analytic functions on the set of probability measures µ with a given finite support.…”

Section: Introductionmentioning

confidence: 99%

Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups

Mathieu,

Sisto

2014

Preprint

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We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be "aligned". We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences including Central Limit Theorems, the local Lipschitz continuity of the rate of escape and entropy, as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point. In a second part of the paper, we show that the (exponential) deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups. These include nonelementary (relatively) hyperbolic groups, Mapping Class Groups, many groups acting on CAT(0) spaces and small cancellation groups.

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“…They showed that the conditional and corresponding unconditional SLLN limits coincide. Finally, entropy properties of the range process of random walks on finitely and infinitely generated discrete groups were studied by Chen, Xie and Zhao [8], Erschler [14] and Windsch [30].…”

Section: Introductionmentioning

confidence: 99%

Capacity of the range of random walks on groups

Mrazović¹,

Sandrić²,

Šebek³

2021

Preprint

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In this paper, we discuss asymptotic behavior of the capacity of the range of symmetric simple random walks on finitely generated groups. We show the corresponding strong law of large numbers and central limit theorem. Here, τ +A denotes the first return time of {S n } n≥0 to the set A, i.e. τ + A = inf{n ≥ 1 : S n ∈ A}. We are interested in the asymptotic behavior of the process {C n } n≥0 defined asObserve that when {S n } n≥0 is recurrent then C n ≡ 0.

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On Continuity of Range, Entropy and Drift for Random Walks on Groups

Cited by 9 publications

References 9 publications

Asymptotic Entropy of Random Walks on Free Products

Asymptotic Entropy of Random Walks on Free Products

Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups

Capacity of the range of random walks on groups

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