Random walks in the hyperbolic plane and the question mark function

Abstract

Consider $G=S{L}_2(\mathbb{Z})/\{\pm I\}$ acting on the complex upper half plane $H$ by $h_M(z)=\frac{az+b}{cz+d},$ for $M\in G$. Let $D=\{z\in H:|z|\ge 1,|\Re (z)|\le 1/2\}$. We consider the set $\mathcal{E}\subset G$ with the $9$ elements $M$, different from the identity, such that $(M{M}^T)\le 3$. We equip the tiling of $H$ defined by $\mathbb{D}=\{h_M(D),M\in G\}$ with a graph structure where the neighbours are defined by $h_M(D)\cap h_{M^{\prime }}(D)\ne \varnothing$, equivalently $M^{-1}{M}^{\prime }\in \mathcal{E}$. The present paper studies several Markov chains related to the above structure. We show that the simple random walk on the above graph converges a.s. to a point $X$ of the real line with the same distribution of $S_2{W}^{S_1}$, where $S_1,S_2,W$ are independent with $\mathrm{Pr}(S_i=\pm 1)=1/2$ and where $W$ is valued in $(0,1)$ with distribution $\mathrm{Pr}(W<w)=?(w)$. Here $?$ is the Minkowski function. If $K_1,K_2,\dots$ are i.i.d with distribution $\mathrm{Pr}(K_i=n)=1/2^n$ for $n=1,2,\dots$, then $W=\frac{1}{K_1+\frac{1}{K_2+\dots }}$: this known result (Isola (2014)) is derived again here.