Markov chains on hyperbolic-like groups and quasi-isometries

Goldsborough, Antoine; Sisto, Alessandro

doi:10.48550/arxiv.2111.09837

Cited by 2 publications

(2 citation statements)

References 30 publications

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“…In the independent and identically distributed case, the dependence of 𝛼 on 𝜖 has been specified and quantitative estimates have been recently obtained when 𝐻 is proper (see [1,3]). Finally, see also the recent work of Goldsborough-Sisto [37] for another perspective on Markovian random products of isometries.…”

Section: Theorem 46 (Markovian Random Walks On Gromov-hyperbolic Spaces)mentioning

confidence: 99%

Counting and boundary limit theorems for representations of Gromov‐hyperbolic groups

Cantrell

Sert

2023

Proceedings of London Math Soc

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Given a Gromov‐hyperbolic group endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of . More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry–Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates address a question of Kaimanovich–Kapovich–Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson–Sullivan measures on the boundary of the group.

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Section: Theorem 46 (Markovian Random Walks On Gromov-hyperbolic Spaces)mentioning

confidence: 99%

Counting and boundary limit theorems for representations of Gromov‐hyperbolic groups

Cantrell

Sert

2023

Proceedings of London Math Soc

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“…In the iid case, the dependence of α on has been specified and quantitative estimates have been recently obtained when H is proper (see [2,3]). Finally, see also the recent work of Goldsborough-Sisto [39] for another perspective on Markovian random products of isometries.…”

Section: 41mentioning

confidence: 99%

Counting and boundary limit theorems for representations of Gromov-hyperbolic groups

Cantrell¹,

Sert²

2022

Preprint

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Given a Gromov-hyperbolic group G endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of G. More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry-Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates provide a positive answer to a question of Kaimanovich-Kapovich-Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson-Sullivan measures on the boundary of the group.Contents 42 9. On a question of Kaimanovich-Kapovich-Schupp 47 References 49

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Markov chains on hyperbolic-like groups and quasi-isometries

Cited by 2 publications

References 30 publications

Counting and boundary limit theorems for representations of Gromov‐hyperbolic groups

Counting and boundary limit theorems for representations of Gromov‐hyperbolic groups

Counting and boundary limit theorems for representations of Gromov-hyperbolic groups

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