Random walks on stochastic hyperbolic half planar triangulations

TL;DR

This paper investigates the behavior of simple random walks on stochastic hyperbolic half planar triangulations, demonstrating that the walker escapes at positive speed and the return probability decays exponentially with a specific rate.

Contribution

It establishes almost sure escape at positive speed and quantifies the return probability decay for random walks on these complex geometric structures.

Findings

Walker escapes boundary at positive speed

Return probability scales as exp(-cn^{1/3})

Provides insights into random walk behavior on hyperbolic maps

Abstract

We study the simple random walk on stochastic hyperbolic half planar triangulations constructed in Angel and Ray [3]. We show that almost surely the walker escapes the boundary of the map in positive speed and that the return probability to the starting point after n steps scales like $\exp(-cn^{1/3})$.