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The CLT for lamplighter groups with an acylindrically hyperbolic base

Maksym Chaudkhari, Christian Gorski, Eduardo Silva

TL;DR

The paper establishes a Central Limit Theorem for the drift of a random walk on lamplighter groups with base H an acylindrically hyperbolic group and lamp group A, under a finite exponential moment and non-elementary projection to H. The authors link the wreath-product word length to the Traveling Salesman Problem in the base group and derive deterministic bounds on the TSP defect, exploiting geodesic tracking results in acylindrically hyperbolic settings. By adapting Mathieu-Sisto deviation inequalities to a slowly growing defect and proving polynomial-in-log moments of the defect, they obtain a Gaussian limit with variance σ^2>0 and poly-logarithmic moment bounds for the drift, extending the results to arbitrary finitely generated lamp groups. The methods provide a robust framework for CLTs on a broad class of wreath products, highlighting the geometric interplay between base-group hyperbolicity, TSP structure, and random-walk fluctuations, with potential implications for related stochastic processes on groups.

Abstract

We prove a Central Limit Theorem for the drift of a non-elementary random walk with a finite exponential moment on a wreath product $A\wr H=\bigoplus_{H} A\rtimes H$ with $A$ a non-trivial finite group and $H$ a finitely generated acylindrically hyperbolic group. We also provide the upper bounds on the central moments of the drift. Furthermore, our results extend to the case where $A$ is an arbitrary (possibly infinite) finitely generated group.

The CLT for lamplighter groups with an acylindrically hyperbolic base

TL;DR

The paper establishes a Central Limit Theorem for the drift of a random walk on lamplighter groups with base H an acylindrically hyperbolic group and lamp group A, under a finite exponential moment and non-elementary projection to H. The authors link the wreath-product word length to the Traveling Salesman Problem in the base group and derive deterministic bounds on the TSP defect, exploiting geodesic tracking results in acylindrically hyperbolic settings. By adapting Mathieu-Sisto deviation inequalities to a slowly growing defect and proving polynomial-in-log moments of the defect, they obtain a Gaussian limit with variance σ^2>0 and poly-logarithmic moment bounds for the drift, extending the results to arbitrary finitely generated lamp groups. The methods provide a robust framework for CLTs on a broad class of wreath products, highlighting the geometric interplay between base-group hyperbolicity, TSP structure, and random-walk fluctuations, with potential implications for related stochastic processes on groups.

Abstract

We prove a Central Limit Theorem for the drift of a non-elementary random walk with a finite exponential moment on a wreath product with a non-trivial finite group and a finitely generated acylindrically hyperbolic group. We also provide the upper bounds on the central moments of the drift. Furthermore, our results extend to the case where is an arbitrary (possibly infinite) finitely generated group.

Paper Structure

This paper contains 17 sections, 11 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Organization
  4. Acknowledgments
  5. Preliminaries
  6. Metric spaces and paths
  7. The Traveling Salesman Problem.
  8. Acylindrically hyperbolic groups
  9. Wreath products
  10. The standard word metric on wreath products
  11. Random walks on groups
  12. Mathieu-Sisto deviation inequalities and consequences
  13. Defective adapted cocycles and the central limit theorem
  14. Deviation inequalities with slowly growing upper bound
  15. Deterministic upper bounds on the defect of the TSP
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $A$ be a non-trivial finite group and let $H$ be a finitely generated acylindrically hyperbolic group. Let $d$ be a standard metric on $A\wr H$ and let $\mu$ be a symmetric probability measure on $A\wr H$ with a finite exponential moment. Suppose that the projection of $\mu$ to $H$ is a generati converges in distribution to a standard Gaussian. In addition, for any $p > 1$ there exists a const

Figures (2)

  • Figure 1: Hypotheses of Lemma \ref{['comb_lemma_1d']}.
  • Figure 2: Illustration of the proof of Claim \ref{['leap_projection']}.

Theorems & Definitions (31)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Theorem 3.2: MathieuSisto2020
  • Theorem 3.3: MathieuSisto2020
  • ...and 21 more