Cutoff on Hyperbolic Surfaces

TL;DR

This paper investigates how the cutoff phenomenon manifests in random walks on hyperbolic surfaces, showing that under certain spectral conditions, distances concentrate and the walk exhibits cutoff, extending prior graph results to geometric settings.

Contribution

It extends the understanding of cutoff phenomena from Ramanujan graphs to hyperbolic surfaces, using eigenvalue density theorems and without relying on the Selberg conjecture.

Findings

Distances are highly concentrated around minimal values under spectral conditions.

Discrete random walks on hyperbolic surfaces exhibit cutoff behavior.

Results apply to arithmetic lattices without assuming Selberg's conjecture.

Abstract

In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4, then the distances on the surface are highly concentrated around the minimal possible value, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres ([20]) from the setting of Ramanujan graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from [27], we are able to show that the results apply to congruence subgroups of $SL_{2}\left(\mathbb{Z}\right)$ and other arithmetic lattices, without relying on the well known conjecture of Selberg ([28]). Conceptually, we show the close relation between the cutoff phenomenon and temperedness of representations of algebraic groups over local fields, partly answering a question of Diaconis ([7]), who asked under what general phenomena cutoff exists.