Cutoff for geodesic paths on hyperbolic manifolds
Probability
2026-05-06 v2 Geometric Topology
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.
Cite
@article{arxiv.2502.06325,
title = {Cutoff for geodesic paths on hyperbolic manifolds},
author = {Charles Bordenave and Joffrey Mathien},
journal= {arXiv preprint arXiv:2502.06325},
year = {2026}
}
Comments
30 pages. Post-publication corrected version: in Proposition 2.3, the constant c_d is corrected by removal of a factor 1/2. Otherwise the manuscript agrees with the published paper