Stability phenomena for Martin boundaries of relatively hyperbolic groups

Dussaule, Matthieu; Gekhtman, Ilya

doi:10.1007/s00440-020-01000-w

Cited by 5 publications

(29 citation statements)

References 30 publications

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“…This completes the results of [13]. Along Section 3, we prove precise results on the asymptotics of the Green function in a virtually abelian finitely generated group at the spectral radius which will show this stability.…”

supporting

confidence: 60%

“…In Section 4, we assume that parabolic subgroups are virtually abelian and show that the Martin boundary is stable in the sense of Definition 2.6 below (see Theorem 4.1). This had already been shown in [13] when the random walk is spectrally non degenerate; it still holds for convergent (hence spectrally degenerate) random walks.…”

Section: Local Limit Theoremmentioning

confidence: 57%

“…The Martin boundary of any finitely supported symmetric admissible random walk on a hyperbolic group is strongly stable. More generally, the Martin boundary of a finitely supported symmetric and admissible random walk on a relatively hyperbolic group is studied in [16], [14] and [13]. In particular, whenever the parabolic subgroups are virtually abelian, the homeomorphism type of the r-Martin boundary is described in [13].…”

Section: Theorem 21 ([5]mentioning

confidence: 99%

“…More generally, the Martin boundary of a finitely supported symmetric and admissible random walk on a relatively hyperbolic group is studied in [16], [14] and [13]. In particular, whenever the parabolic subgroups are virtually abelian, the homeomorphism type of the r-Martin boundary is described in [13]. It is proved there that the Martin boundary is strongly stable if and only if the random walk is not spectrally degenerate.…”

Section: Theorem 21 ([5]mentioning

confidence: 99%

“…On relatively hyperbolic groups, the first author proved in [11] that the local limit theorem (2) still holds provided the random walk is spectrally non degenerate. This will be precisely defined in Definition 3.1 below, see also [13,Definition 2.3]; roughly speaking, the random walk is spectrally degenerate along a parabolic subgroup H if it gives enough weight to H ; when it is not spectrally along all the parabolic subgroup in Ω, its is called spectrally non-degenerate. The spirit of the result in [11] is that a spectrally non degenerate random walk mainly sees the underlying hyperbolic structure of the group.…”

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confidence: 99%

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A local limit theorem for convergent random walks on relatively hyperbolic groups

Dussaule¹,

Peigné²,

Tapie³

2022

Preprint

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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 pn(e, e) ∼ CR −n n −d/2 , where pn(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 classification all possible behaviour for pn(e, e) on such groups.

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supporting

confidence: 60%

Section: Local Limit Theoremmentioning

confidence: 57%

Section: Theorem 21 ([5]mentioning

confidence: 99%

Section: Theorem 21 ([5]mentioning

confidence: 99%

mentioning

confidence: 99%

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A local limit theorem for convergent random walks on relatively hyperbolic groups

Dussaule¹,

Peigné²,

Tapie³

2022

Preprint

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Local limit theorems in relatively hyperbolic groups I: rough estimates

Dussaule

2021

Ergod. Th. Dynam. Sys.

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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 hyperbolic 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, where $p_n$ is the probability of going back to the origin at time n and where R is the inverse of the spectral radius of the random walk.

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Local limit theorems in relatively hyperbolic groups II: the non-spectrally degenerate case

Dussaule¹

2022

Compositio Math.

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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 part I to show that $p_n(e,e)\sim CR^{-n}n^{-3/2}$ , where $R$ is the inverse of the spectral radius of the random walk. This both generalizes results of Woess for free products and results of Gouëzel for hyperbolic groups.

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Stability phenomena for Martin boundaries of relatively hyperbolic groups

Cited by 5 publications

References 30 publications

A local limit theorem for convergent random walks on relatively hyperbolic groups

A local limit theorem for convergent random walks on relatively hyperbolic groups

Local limit theorems in relatively hyperbolic groups I: rough estimates

Local limit theorems in relatively hyperbolic groups II: the non-spectrally degenerate case

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