Central limit theorems for counting measures in coarse negative curvature

Gekhtman, Ilya; Taylor, Selwyn; Tiozzo, Giulio

doi:10.48550/arxiv.2004.13084

Cited by 4 publications

(19 citation statements)

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“…The core of the argument is based on large deviation estimates from Benoist-Quint [8] that we develop (Theorem 3.3) for the Markovian setting by elaborating on other work due to Benoist-Quint [6] which concerns large deviation estimates for cocycles. Equipped with these results as well as techniques developed by Calegari-Fujiwara [18], we employ a quantitative version of an argument from recent work of Gekhtman-Taylor-Tiozzo [34] to carry out our proof. 1.2.3.…”

Section: Positivity Of Average Growth Ratementioning

confidence: 99%

“…The following result establishes exponential counting large deviation estimates in another setting, that of isometric actions on Gromov-hyperbolic spaces. This setting has recently attracted much attention both from probabilistic [2,4,7,14,40,53] and counting [18,22,23,24,34,35,68] perspectives. To state our result, recall that the action of a group Γ on a Gromov-hyperbolic space H by isometries is said to be nonelementary if it there exists γ 1 , γ 2 ∈ Γ acting as loxodromic elements (see §3.5) with disjoint pairs of fixed points on the Gromov boundary of H.…”

Section: Positivity Of Average Growth Ratementioning

confidence: 99%

“…Coming back to counting asymptotics, there has recently been significant interest in counting limit theorems on hyperbolic groups, see for example [18,22,24,26,34,35,45,46,57,68]. In some of these works, techniques from thermodynamic formalism (tracing back to [15,64], see also [50]) are used.…”

Section: Previous Workmentioning

confidence: 99%

“…In others, including ours, ideas from Markov chain or random walk theory (in a sense initiated in this context by [18]) are used instead of thermodynamic techniques. For example, in [34] the authors deduce a (qualitative) counting CLT for displacement functions on Gromov-hyperbolic spaces 2 from a CLT for centerable cocycles. Using ideas of Benoist-Quint [6] relying on solving a cohomological equation, it might be possible to do so in our setting as well.…”

Section: Previous Workmentioning

confidence: 99%

“…For example, for the last three results mentioned above, it enables us to employ probabilistic results of Markovian random matrix products (mainly due to Bougerol [11,12,13] and Guivarc'h [43]; we also develop some of them further) to the deterministic counting results. This transfer, however, requires handling some difficulties which we manage to do by, among others, elaborating on techniques developed by Calegari-Fujiwara [18] (generally) and Gekhtman-Taylor-Tiozzo [34] (for the central limit theorem). The deterministic nature of our results, in particular the fact that we do not induce randomness using an external source (like a subshift of finite type [18,58]) is of particular interest.…”

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confidence: 99%

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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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Section: Positivity Of Average Growth Ratementioning

confidence: 99%

Section: Positivity Of Average Growth Ratementioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

mentioning

confidence: 99%

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

Cantrell¹,

Sert²

2022

Preprint

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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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Central limit theorem and geodesic tracking on hyperbolic spaces and Teichmüller spaces

Choi¹

2021

Preprint

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We study random walks on the isometry group of a Gromov hyperbolic space or Teichmüller space. We prove that the translation lengths of random isometries satisfy a central limit theorem if and only if the random walk has finite second moment. We also recover the central limit theorem of Benoist and Quint for the displacement of a reference point and establish its converse. We also prove sublinear geodesic tracking for random walks with finite (1/2)-th moment and sublogarithmic geodesic tracking of random walks with finite exponential moment.

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Central limit theorems for counting measures in coarse negative curvature

Cited by 4 publications

References 0 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

Central limit theorem and geodesic tracking on hyperbolic spaces and Teichmüller spaces

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