Cutoff for geodesic paths on hyperbolic manifolds

TL;DR

This paper demonstrates the cutoff phenomenon for geodesic paths and Brownian motion on compact hyperbolic manifolds, extending previous results to higher dimensions using spectral analysis techniques.

Contribution

It establishes the cutoff phenomenon for geodesic paths on all compact hyperbolic manifolds, generalizing prior work on hyperbolic surfaces to any dimension.

Findings

Proves cutoff for geodesic paths on compact hyperbolic manifolds.

Extends results from hyperbolic surfaces to higher dimensions.

Uses spectral analysis of the spherical mean operator.

Abstract

We establish new instances of the cutoff phenomenon for geodesic paths and for the Brownian motion on compact hyperbolic manifolds. We prove that for any fixed compact hyperbolic manifold, the geodesic path started on a spatially localized initial condition exhibits cutoff. Our work also extends results obtained by Golubev and Kamber on hyperbolic surfaces of large volume to any dimension. Our proof builds upon a spectral strategy introduced by Lubetzky and Peres for Ramanujan graphs and on a detailed spectral analysis of the spherical mean operator.