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Random walks and contracting elements I: Deviation inequality and Limit laws

Inhyeok Choi

TL;DR

This work develops a unified, boundary-free theory of random walks on metric spaces with contracting elements, introducing pivotal times and Schottky sets to obtain sharp deviation bounds and limit laws without relying on global hyperbolicity. By blending Gouëzel’s pivotal-time construction with Sisto’s contracting-axes framework, the authors prove exponential deviation bounds and then derive CLTs, laws of the iterated logarithm, and geodesic-tracking results for mapping class groups and rank-1 CAT(0) spaces under optimal moment conditions. The paper also establishes a large deviation principle and extends the framework to weakly contracting directions and HHGs, including mapping class groups, thereby providing a versatile toolkit for limit theorems in non-hyperbolic settings. The approaches generalize prior hyperbolic-space results and yield quantitative, probabilistic descriptions of random-walk escape, alignment, and path geometry in these rich geometric contexts.

Abstract

We study random walks on metric spaces with contracting isometries. In this first article of the series, we establish sharp deviation inequalities by adapting Gouëzel's pivotal time construction. As an application, we establish the exponential bounds for deviation from below, central limit theorem, law of the iterated logarithms and the geodesic tracking of random walks on mapping class groups and CAT(0) spaces.

Random walks and contracting elements I: Deviation inequality and Limit laws

TL;DR

This work develops a unified, boundary-free theory of random walks on metric spaces with contracting elements, introducing pivotal times and Schottky sets to obtain sharp deviation bounds and limit laws without relying on global hyperbolicity. By blending Gouëzel’s pivotal-time construction with Sisto’s contracting-axes framework, the authors prove exponential deviation bounds and then derive CLTs, laws of the iterated logarithm, and geodesic-tracking results for mapping class groups and rank-1 CAT(0) spaces under optimal moment conditions. The paper also establishes a large deviation principle and extends the framework to weakly contracting directions and HHGs, including mapping class groups, thereby providing a versatile toolkit for limit theorems in non-hyperbolic settings. The approaches generalize prior hyperbolic-space results and yield quantitative, probabilistic descriptions of random-walk escape, alignment, and path geometry in these rich geometric contexts.

Abstract

We study random walks on metric spaces with contracting isometries. In this first article of the series, we establish sharp deviation inequalities by adapting Gouëzel's pivotal time construction. As an application, we establish the exponential bounds for deviation from below, central limit theorem, law of the iterated logarithms and the geodesic tracking of random walks on mapping class groups and CAT(0) spaces.
Paper Structure (26 sections, 3 theorems, 313 equations, 4 figures)

This paper contains 26 sections, 3 theorems, 313 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Strategy
  4. Preliminaries
  5. Paths
  6. Strong contraction
  7. Weak contraction
  8. Random walks
  9. Random walks with strongly contracting isometries
  10. Alignment I: strongly contracting axes
  11. Contracting geodesics
  12. Alignment
  13. Schottky sets
  14. Pivoting and limit laws
  15. Pivotal times: statement
  16. ...and 11 more sections

Key Result

Theorem A

Let $G$ be the mapping class group of a finite-type surface, let $d$ be a word metric on $G$, let $(Z_{n})_{n\ge0}$ be the random walk generated by a non-elementary probability measure $\mu$ on $G$, and let be the drift of $\mu$ on $G$. Then for each $0 < L < \lambda$, the probability $\mathop{\mathrm{\mathbb{P}}}\nolimits (d(id, Z_{n}) \le Ln)$ decays exponentially as $n$ tends to infinity.

Figures (4)

  • Figure 1: Axes associated with a sequence of isometries $s = (\phi_{1}, \phi_{2}, \phi_{3}, \phi_{4})$. Points inside the darker shadow constitute $\Gamma^{+}(s)$, and those inside the lighter shadow constitute $\Gamma^{2}(s)$. Points in the dashed region constitute $\Gamma^{-}(s)$.
  • Figure 2: Schematics for an aligned sequence of paths.
  • Figure 3: Persistent progress and $\mathop{\mathrm{\upsilon}}\nolimits$. Here, all of the backward loci $(\check{Z}_{n} o)_{n \ge 0}$ are on the left of the persistent progress $Z_{i}\Gamma^{+}(\alpha)$, while the forward loci after $Z_{\mathop{\mathrm{\varsigma}}\nolimits} o$ are all on the right.
  • Figure 4: Schematics for Criteria \ref{['eqn:pivotingCond1']}, \ref{['eqn:pivotingCond2']}, \ref{['eqn:pivotingCond3']} and \ref{['eqn:pivotingCond4']}.

Theorems & Definitions (57)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.5
  • Definition 2.6
  • Definition 2.8
  • proof
  • proof
  • proof
  • ...and 47 more