Speed Exponents of Random Walks on Groups

Amir, Gideon; Virág, Bálint

doi:10.1093/imrn/rnv378

Cited by 12 publications

(30 citation statements)

References 20 publications

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“…In particular they contain as subgroups all groups generated by finite-state automata with bounded activity (see Theorem 5.2 below). A particular case of the group M (A, B) was also used in [Bri13,AV12].…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…The following proposition determines a lower bound for the asymptotics of resistance in Λ n between the root and any anti-root, as n → ∞. See also [AV14,AV12] for more on resistances in these graphs.…”

Section: Resistances In Schreier Graphsmentioning

confidence: 96%

“…(For the definition and preliminaries on the Liouville property see Section 1.2.) On one hand, construction based on these groups have provided new examples of asymptotic behaviours for the rate of escape and entropy of random walks in the sublinear range [Bri13,AV12] and for the relationship between the Liouville property and growth of groups [Ers04a,BE11]. On the other hand the Liouville property has turned out to be a useful tool to prove amenability of several classes of groups acting on rooted trees [BV05,Kai05,Bri09,BKN10,AAV13].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Liouville property for groups acting on rooted trees

Amir¹,

Angel²,

Bon³

et al. 2016

Ann. Inst. H. Poincaré Probab. Statist.

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Section: Statement Of Resultsmentioning

confidence: 99%

Section: Resistances In Schreier Graphsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Liouville property for groups acting on rooted trees

Amir¹,

Angel²,

Bon³

et al. 2016

Ann. Inst. H. Poincaré Probab. Statist.

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“…The inverted orbit is a well-known object, central to the study of growth and random walks on permutational wreath products, see [BE12,BE11,AV12]. It is sometimes convenient to consider the following variation of the inverted orbit…”

Section: Probabilistic Reformulation and Recurrent Actions 41 The Inmentioning

confidence: 99%

Extensive amenability and an application to interval exchanges

Juschenko¹,

Bon

Monod

2016

Ergod. Th. Dynam. Sys.

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Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank ≤ 2.In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on IET and show that there are subgroups G < IET admitting no finitely supported measure with trivial boundary.

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“…The following lemma generalizes [6,Proposition 4.11]; see also [1,9]: Lemma 6.2. Let pG i q be a self-similar sequence of groups, and let pµ i q be a selfsimilar sequence of measures on pG i q, with laziness pα i q.…”

mentioning

confidence: 93%

Poisson–Furstenberg boundary and growth of groups

Bartholdi

Erschler

2016

Probab. Theory Relat. Fields

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Abstract. We study the Poisson-Furstenberg boundary of random walks on permutational wreath products. We give a sufficient condition for a group to admit a symmetric measure of finite first moment with non-trivial boundary, and show that this criterion is useful to establish exponential word growth of groups. We construct groups of exponential growth such that all finitely supported (not necessarily symmetric, possibly degenerate) random walks on these groups have trivial boundary. This gives a negative answer to a question of Kaimanovich and Vershik.

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Speed Exponents of Random Walks on Groups

Abstract: For every 3/4 ≤ β < 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point satisfies E|X n | ≍ n β . In fact, the speed can be set precisely to equal any nice prescribed function up to a constant factor.

Cited by 12 publications

References 20 publications

The Liouville property for groups acting on rooted trees

The Liouville property for groups acting on rooted trees

Extensive amenability and an application to interval exchanges

Poisson–Furstenberg boundary and growth of groups

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