Random walk on surfaces with hyperbolic cusps

TL;DR

This paper analyzes the spectral properties of a random walk on hyperbolic cusp surfaces, providing bounds on the spectral gap and insights into the convergence rate to the stationary distribution.

Contribution

It introduces a spectral analysis framework for random walks on hyperbolic cusp surfaces, deriving bounds on the spectral gap and convergence rates.

Findings

Established bounds on the spectral gap of the operator

Quantified the convergence rate of the random walk to stationarity

Provided spectral analysis techniques for hyperbolic cusp surfaces

Abstract

We consider the operator associated to a random walk on finite volume surfaces with hyperbolic cusps. We study the spectral gap (upper and lower bound) associated to this operator and deduce some rate of convergence of the iterated kernel towards its stationary distribution.