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

Inhyeok Choi

TL;DR

This work develops a comprehensive probabilistic-geometric framework to study random walks on isometries of Gromov hyperbolic spaces and Teichmüller space, proving that the translation length $ au(g)$ satisfies a central limit theorem if and only if the step distribution has finite second moment, while the displacement CLT holds under Benoist–Quint-type conditions and its converse is established. The authors introduce pivoting and Schottky-set techniques to control alignment and deviations, enabling central limit and iterated logarithm laws even in non-proper settings and within Teichmüller space. They also prove sublinear and logarithmic geodesic tracking under moment conditions and provide a robust, non-martingale-based route to Gaussian limits via uniform deviation inequalities. The results extend the known CLTs from proper hyperbolic settings to Teichmüller space and non-proper spaces, with potential implications for mapping class groups and related spaces through cross-space dynamics.

Abstract

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. While doing this, we recover the central limit theorem of Benoist and Quint for the displacement of a reference point and establish its converse. Also discussed are the corresponding laws of the iterated logarithm. Finally, we prove sublinear geodesic tracking by random walks with finite $(1/2)$-th moment and logarithmic tracking by random walks with finite exponential moment.

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

TL;DR

This work develops a comprehensive probabilistic-geometric framework to study random walks on isometries of Gromov hyperbolic spaces and Teichmüller space, proving that the translation length satisfies a central limit theorem if and only if the step distribution has finite second moment, while the displacement CLT holds under Benoist–Quint-type conditions and its converse is established. The authors introduce pivoting and Schottky-set techniques to control alignment and deviations, enabling central limit and iterated logarithm laws even in non-proper settings and within Teichmüller space. They also prove sublinear and logarithmic geodesic tracking under moment conditions and provide a robust, non-martingale-based route to Gaussian limits via uniform deviation inequalities. The results extend the known CLTs from proper hyperbolic settings to Teichmüller space and non-proper spaces, with potential implications for mapping class groups and related spaces through cross-space dynamics.

Abstract

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. While doing this, we recover the central limit theorem of Benoist and Quint for the displacement of a reference point and establish its converse. Also discussed are the corresponding laws of the iterated logarithm. Finally, we prove sublinear geodesic tracking by random walks with finite -th moment and logarithmic tracking by random walks with finite exponential moment.

Paper Structure

This paper contains 17 sections, 4 theorems, 241 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gromov hyperbolic spaces
  4. Teichmuller space
  5. Random walks
  6. Witnessing and alignment
  7. Pivotal times
  8. Schottky sets and pivots
  9. Pivotal times
  10. Pivots in random walks
  11. The first model and pivoting
  12. Pivoting and its consequences
  13. Central limit theorems
  14. Converse of central limit theorems
  15. Central limit theorems
  16. ...and 2 more sections

Key Result

Theorem A

Suppose that $\mu$ has finite first moment. Then there exists a constant $K < \infty$ such that for almost every $\mathop{\mathrm{\omega}}\nolimits$.

Figures (5)

  • Figure 1: $D$-witnessing in Teichmüller space. Here $[x, y]$ is $D$-witnessed by $([x_{1}, y_{1}], [x_{2}, y_{2}], [x_{3}, y_{3}])$.
  • Figure 2: Schematics for Lemma \ref{['lem:1segment']}, \ref{['lem:concat']} and \ref{['lem:concatUlt']}.
  • Figure 3: Alignment and marking. Here, $\left(\gamma_{i}\right)_{i=1}^{4}$ and $\left(\eta_{i}\right)_{i=1}^{4}$ are $D$-aligned and $[x, y]$ is $(C, D)$-marked with $\left(\gamma_{i}\right)_{i=1}^{4}$, $\left(\eta_{i}\right)_{i=1}^{4}$. We also say that $[p_{1}, y]$ is $(C, D)$-head-marked with $\left(\gamma_{i}\right)_{i=1}^{4}$ and $\left(\eta_{i}\right)_{i=2}^{4}$. Similarly, we say that $[x, p_{4}]$ is $(C, D)$-tail-marked with $\left(\gamma_{i}\right)_{i=1}^{3}$, $\left(\eta_{i}\right)_{i=1}^{4}$.
  • Figure 4: Loci $y_{i, k}^{\pm}$ inside a trajectory.
  • Figure 5: $\{Y_{k, i}\}$, $\{Y_{k;n}\}$, $\{b_{k, i}\}$ and $\{b_{k;n}\}$ for $10 \cdot 2^{m} \le n \le 11 \cdot 2^{m}$. Here $b_{0;n} = b_{2;n} = 0$ since $2^{m} (2 \lfloor n/2^{m+1} \rfloor + 1) = 11 \cdot 2^{m} \ge n$ and $2^{m+2} (2 \lfloor n/2^{m+3} \rfloor + 1) = 12 \cdot 2^{m} \ge n$.

Theorems & Definitions (49)

  • Theorem A: Logarithmic deviation
  • Theorem B: Central limit theorems and Laws of the Iterated Logarithm
  • Theorem C: Converse of Central limit theorems
  • Theorem D: Geodesic tracking
  • Remark 1.1
  • Definition 2.1
  • Definition 2.3
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 39 more