Cutoff for geodesic paths on hyperbolic manifolds

TL;DR

This work proves cutoff phenomena for geodesic paths and Brownian motion on compact hyperbolic manifolds, establishing abrupt convergence to the uniform (volume) measure from localized initial data. It extends Lubetzky–Peres type spectral strategies to hyperbolic geometry via detailed analysis of the spherical mean operator $A_t$, linking mixing times to spectral edge behavior and, in particular, to near-optimal gaps or the Sarnak-Xue density property. The results provide explicit cutoff times, $t_* = \ln V_n/(d-1)$ (and $2t_*/(d-1)$ for Brownian motion), with lower bounds from volume growth and upper bounds from refined spectral bounds, including for high-dimensional manifolds. The findings highlight a robust cutoff mechanism in negatively curved spaces and open avenues to universal behavior in similar dynamical systems and other flows or processes on manifolds. These insights advance the understanding of mixing in geometric contexts and potentially inform further exploration of cutoff in non-constant curvature or chaotic billiards.

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.

Theorems & Definitions (17)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Proposition 2.1
• Proposition 2.2
• proof
• proof : Proof of Property \ref{['prop:LB']}