Reduced Random Walks in the Hyperbolic Plane$!?$

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 $PG{L}_2(\mathbb{Z})$ (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 $!?:[0,1]\to \mathbb{R}$ that shares several of the remarkable analytic and arithmetic properties of Minkowski's question-mark function.

26 pages