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Reduced Random Walks in the Hyperbolic Plane$\hspace{1pt}!\hspace{-3.8pt}?$

Colin Defant, Mitchell Lee

TL;DR

This work analyzes Lam's reduced random walk on hyperbolic triangle groups by embedding the walk in the hyperbolic plane and proving almost sure convergence of the geometric trajectory to a boundary point. It establishes a unique stationary distribution on the boundary characterized by a simple distributional equation involving one-way reflections, and specializes to $W \simeq \mathrm{PGL}_2(\mathbb{Z})$ to obtain an explicit cumulative distribution function for the limit via the interrobang function. The interrobang function, defined on rationals by a height-based recursion and extending continuously to $[0,1]$, mirrors many properties of Minkowski's question-mark function and drives the explicit description of the limit distribution, connecting hyperbolic random walks with arithmetic dynamics. The results provide a concrete framework for boundary behavior of reduced random walks on hyperbolic groups and introduce new analytic/arithmetic tools with potential applications in number theory and dynamics.

Abstract

We study Lam's reduced random walk in a hyperbolic triangle group, which we view as a random walk in the upper half-plane. We prove that this walk converges almost surely to a point on the extended real line. We devote special attention to the reduced random walk in $PGL_2(\mathbb{Z})$ (i.e., the $(2,3,\infty)$ triangle group). In this case, we provide an explicit formula for the cumulative distribution function of the limit. This formula is written in terms of the interrobang function, a new function $!\hspace{-3.8pt}?\colon[0,1]\to\mathbb{R}$ that shares several of the remarkable analytic and arithmetic properties of Minkowski's question-mark function.

Reduced Random Walks in the Hyperbolic Plane$\hspace{1pt}!\hspace{-3.8pt}?$

TL;DR

This work analyzes Lam's reduced random walk on hyperbolic triangle groups by embedding the walk in the hyperbolic plane and proving almost sure convergence of the geometric trajectory to a boundary point. It establishes a unique stationary distribution on the boundary characterized by a simple distributional equation involving one-way reflections, and specializes to to obtain an explicit cumulative distribution function for the limit via the interrobang function. The interrobang function, defined on rationals by a height-based recursion and extending continuously to , mirrors many properties of Minkowski's question-mark function and drives the explicit description of the limit distribution, connecting hyperbolic random walks with arithmetic dynamics. The results provide a concrete framework for boundary behavior of reduced random walks on hyperbolic groups and introduce new analytic/arithmetic tools with potential applications in number theory and dynamics.

Abstract

We study Lam's reduced random walk in a hyperbolic triangle group, which we view as a random walk in the upper half-plane. We prove that this walk converges almost surely to a point on the extended real line. We devote special attention to the reduced random walk in (i.e., the triangle group). In this case, we provide an explicit formula for the cumulative distribution function of the limit. This formula is written in terms of the interrobang function, a new function that shares several of the remarkable analytic and arithmetic properties of Minkowski's question-mark function.
Paper Structure (19 sections, 19 theorems, 94 equations, 6 figures)

This paper contains 19 sections, 19 theorems, 94 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Random walks on hyperbolic triangle groups
  3. Previous research on reduced random walks
  4. Results
  5. Outline
  6. The reduced random walk on a Coxeter group
  7. Hyperbolic triangle groups
  8. The hyperbolic plane
  9. Hyperbolic triangle groups
  10. Reduced random walks in the hyperbolic plane
  11. Uniqueness of the stationary measure
  12. The Poincaré disk model
  13. The contraction properties
  14. A worked example: PGL₂(ℤ)
  15. The interrobang function
  16. ...and 4 more sections

Key Result

Lemma 4.1

For each line $L \in \mathcal{H}$, there exists an integer $N$ such that the set ${\{z_m \, | \, m \geq N\}}$ lies entirely on one side of the line $L$.

Figures (6)

  • Figure 1: A trajectory of the reduced random walk in $PGL_2(\mathbb{Z})$. The limit point $\zeta\in\overline{\mathbb{R}}$ is labeled.
  • Figure 2: An example of lines $L_1, L_2, L_3 \subseteq \mathbb{H}$ and a point $z_0 \in \mathbb{H}$ satisfying \ref{['ax:angles']}. In this example, $m(1, 2) = 2$ and $m(2, 3) = m(1, 3) = \infty$.
  • Figure 3: An illustration of the first part of the proof of \ref{['lem:strict-contract']}. For $i = 1, 2, 3$, we have drawn the open subset $\{x \in S^1 \, | \, |x - c_i| > r_i / C\} \subseteq S^1$ in red, blue, and green respectively. If $C < 1$ is sufficiently close to $1$, then these sets form an open cover of $S^1$.
  • Figure 4: The transition diagram of the reduced random walk in $PGL_2(\mathbb{Z})$.
  • Figure 5: A plot of the interrobang function.
  • ...and 1 more figures

Theorems & Definitions (45)

  • Lemma 4.1
  • Theorem 4.2
  • proof
  • Remark 4.3
  • Theorem 4.4
  • proof
  • Theorem 5.1
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • ...and 35 more