Random walk on Bethe lattice and hyperbolic geometry

TL;DR

This paper provides an exact solution for a random walk on the Bethe lattice, explores its continuous limit, and discusses its relation to Brownian motion in hyperbolic geometry, revealing insights into stochastic processes on curved spaces.

Contribution

It introduces an exact solution for the random walk on the Bethe lattice and connects it to Brownian motion in hyperbolic space, bridging discrete and continuous models.

Findings

Exact solution for random walk on Bethe lattice

Relation between Bethe lattice walk and hyperbolic Brownian motion

Insights into stochastic processes in negatively curved spaces

Abstract

We give the exact solution to the problem of a random walk on the Bethe lattice through a mapping on an asymmetric random walk on the half-line. We also study the continuous limit of this model, and discuss in detail the relation between the random walk on the Bethe lattice and Brownian motion on a space of constant negative curvature.