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Noise stability on hyperbolic groups

Timothée Bénard, Ryokichi Tanaka

TL;DR

The paper proves noise stability with linear scale for symmetric random walks on a class of non-elementary word hyperbolic groups that admit a nonzero homomorphism to $\mathbb{R}$, under finite exponential moment. It introduces the coupling $\pi^{\rho}$ and shows that for all $\rho\in[0,1)$ the $n$-step distribution $\pi^{\rho}_n$ remains asymptotically separated from the product measure $\mu_n\otimes\mu_n$ in total variation, establishing stability rather than sensitivity. Central to the argument is the singularity of harmonic measures on the boundary product $(\partial\Gamma)^2$ for different $\rho$, obtained via winding statistics and a joint law of the iterated logarithm (LIL) for geodesic projections, together with stopping-time controls and a sharp marginal analysis. These results yield a robust form of stability at linear scale and illuminate the structure of boundary measures for driven random walks on hyperbolic groups.

Abstract

We show that symmetric random walks on non-elementary hyperbolic groups with non-zero homomorphisms into the reals are noise stable at linear scale under finite exponential moment condition.

Noise stability on hyperbolic groups

TL;DR

The paper proves noise stability with linear scale for symmetric random walks on a class of non-elementary word hyperbolic groups that admit a nonzero homomorphism to , under finite exponential moment. It introduces the coupling and shows that for all the -step distribution remains asymptotically separated from the product measure in total variation, establishing stability rather than sensitivity. Central to the argument is the singularity of harmonic measures on the boundary product for different , obtained via winding statistics and a joint law of the iterated logarithm (LIL) for geodesic projections, together with stopping-time controls and a sharp marginal analysis. These results yield a robust form of stability at linear scale and illuminate the structure of boundary measures for driven random walks on hyperbolic groups.

Abstract

We show that symmetric random walks on non-elementary hyperbolic groups with non-zero homomorphisms into the reals are noise stable at linear scale under finite exponential moment condition.
Paper Structure (6 sections, 13 theorems, 44 equations)

This paper contains 6 sections, 13 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. On the proof
  3. Singularity of harmonic measures and winding statistics
  4. Preliminaries and limit laws on single hyperbolic groups
  5. Stopping times and technical lemmas
  6. A joint LIL and singularity of harmonic measures

Key Result

Theorem 1.1

Let $\Gamma$ be a non-elementary word hyperbolic group which admits a non-zero homomorphism to ${\mathbb R}$. Let $\mu$ be a symmetric probability measure on $\Gamma$ with finite exponential moment and whose support is not included in a proper subgroup of $\Gamma$. It holds that for every $\rho \in i.e. the $\mu$-random walk on $\Gamma$ is noise stable in total variation.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Theorem 1.4 in BenardWinding
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 8 more