Random walks reaching against all odds the other side of the quarter plane

TL;DR

This paper derives exact and asymptotic probabilities for a random walk in the quarter plane reaching one axis before the other, with applications in models like nucleosome shifting and voter dynamics.

Contribution

It provides a novel exact integral formula and asymptotic analysis for hitting probabilities in quarter plane random walks, extending understanding of such stochastic processes.

Findings

Exact probability expression involving conformal gluing functions

Asymptotic probability for large initial states

Applications to nucleosome shifting and voter models

Abstract

For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state $(i_0,j_0)$, we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive an exact expression for the probability of this event, and derive an asymptotic expression for the case when $i_0$ becomes large, a situation in which the event becomes highly unlikely. The exact expression follows from the solution of a boundary value problem and is in terms of an integral that involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model and the asymmetric exclusion process.