Boundary contacts for reflected random walks in the quarter plane
par
E2 1180
Tours
Reflected random walks in the quarter plane arise naturally in probability theory, queueing systems, statistical physics, and combinatorics. In this talk, I will discuss the local time spent on the reflection boundaries. When the drift lies within the cone, the local time converges, without normalization, to a nontrivial random variable as the walk length tends to infinity. I will present recent results on these limiting variables in two different settings: in the first, the reflections on the horizontal and vertical boundaries are assumed to be similar, leading to a recursive structure for the probability mass functions that can be analyzed using a coupling approach; in the second, we consider more general reflection rules but singular random walks, for which we derive an explicit closed-form expression for the limiting distribution using the compensation approach. Some perspectives and simple examples on both singular and non-singular models will also be discussed, time permitting. This is joint work with Kilian Rachel.