Quadratic differentials and random walks on the dual graph of a pants decomposition

TL;DR

This paper establishes a connection between the ergodicity of geodesic flows on infinite Riemann surfaces and the recurrence of associated random walks on dual graphs, providing explicit criteria based on cuff-length growth.

Contribution

It introduces a novel equivalence between geodesic flow ergodicity and random walk recurrence on dual graphs, with criteria based on cuff-length growth and phase transition analysis.

Findings

Geodesic flow ergodicity is equivalent to random walk recurrence on the dual graph.

Explicit criteria for ergodicity based on cuff-length growth are provided.

Rough isometry of surfaces does not necessarily preserve ergodicity, but rough isometry of dual graphs does.

Abstract

Let X be an infinite Riemann surface with an upper-bounded geodesic pants decomposition. The vertices of the corresponding dual graph G are pairs of pants and edges are cuffs with conductances equal to their lengths. We prove that the geodesic flow on X is ergodic if and only if the random walk on G is recurrent. This yields explicit criteria for deciding, in terms of cuff-length growth, whether the geodesic flow is ergodic. We provide concrete and new families of Riemann surfaces with an explicit understanding of the phase transitions from recurrent to non-recurrent geodesic flows. In addition, we show that rough isometry of surfaces does not preserve the ergodicity of the geodesic flow while rough isometry of their dual graphs does. The above equivalence result uses a characterization of the measured geodesic laminations on X that arise as straightened horizontal foliations of finite-area holomorphic quadratic differentials. The conditions on the measured laminations are translated into the conditions on the existence of a square summable flow function on G.