Boundary contacts for reflected random walks in the quarter plane

TL;DR

This paper studies the long-term behavior of reflected random walks in the quarter plane, focusing on boundary interactions and deriving explicit distributions under different reflection assumptions.

Contribution

It introduces new recursive and explicit formulas for the limiting boundary local times of reflected random walks with various reflection rules.

Findings

Limiting local time converges without normalization for walks with drift in the cone.

Recursive structure identified for similar boundary reflections via coupling.

Explicit distribution formulas derived for general reflection rules using the compensation approach.

Abstract

We investigate reflected random walks in the quarter plane, with particular emphasis on the time spent along the reflection boundary axes. Assuming the drift of the random walk lies within the cone, the local time converges -- without the need for normalization -- to a limiting random variable as the walk length tends to infinity. This paper focuses on the properties of these discrete limiting variables. The problem is rooted in probability theory but also has natural connections to statistical physics and analytic combinatorics. We present two main sets of results, each based on different assumptions regarding the random walk parameters. First, when the reflections on the horizontal and vertical boundaries are assumed to be similar, we reveal the recursive structure of the problem through a coupling approach. Second, in the case of more general reflection rules but singular random walks, we derive an explicit closed-form expression for the limiting distribution using the compensation approach. Our results are illustrated via concrete computations on various examples.