Nonintersecting random walks in the neighborhood of a symmetric tacnode

TL;DR

This paper studies a model of nonintersecting random walks near a symmetric tacnode, deriving a new limiting kernel that describes their behavior under specific scaling conditions.

Contribution

It introduces a new determinantal process for nonintersecting random walks near a symmetric tacnode and derives the associated extended kernel under a particular scaling.

Findings

Derived the Fredholm determinant for gap probabilities.

Identified the limit extended kernel under the scaling m=2t+σt^{1/3}.

Characterized the interaction strength between the two groups of walks.

Abstract

Consider a continuous time random walk in $\mathbb{Z}$ with independent and exponentially distributed jumps $\pm1$. The model in this paper consists in an infinite number of such random walks starting from the complement of $\{-m,-m+1,\ldots,m-1,m\}$ at time -t, returning to the same starting positions at time t, and conditioned not to intersect. This yields a determinantal process, whose gap probabilities are given by the Fredholm determinant of a kernel. Thus this model consists of two groups of random walks, which are contained within two ellipses which, with the choice $m\simeq2t$ to leading order, just touch: so we have a tacnode. We determine the new limit extended kernel under the scaling $m=\lfloor2t+\sigma t^{1/3}\rfloor$, where parameter $\sigma$ controls the strength of interaction between the two groups of random walkers.