On the exit time from a cone for random walks with drift

TL;DR

This paper analyzes the exponential decay rate of the probability that a multi-dimensional random walk with drift remains within a convex cone over time, linking it to the Laplace transform of the walk's increments and applying it to combinatorial enumeration problems.

Contribution

It provides a precise formula for the decay rate of cone-hitting probabilities for random walks with drift, connecting probabilistic and combinatorial methods.

Findings

Decay rate equals the minimum of the Laplace transform over the dual cone.

Results apply to counting lattice walks in orthants.

Establishes a link between large deviations and combinatorics.

Abstract

We compute the exponential decay of the probability that a given multi-dimensional random walk stays in a convex cone up to time $n$, as $n$ goes to infinity. We show that the latter equals the minimum, on the dual cone, of the Laplace transform of the random walk increments. As an example, our results find applications in the counting of walks in orthants, a classical domain in enumerative combinatorics.