Scaling limit of the collision measures of multiple random walks

TL;DR

This paper investigates the scaling limit of collision measures of multiple independent random walks, showing convergence to a non-trivial random measure characterized via Wiener chaos, inspired by statistical mechanics methods.

Contribution

It establishes the convergence of collision measures of multiple random walks to a limit measure and characterizes this limit using Wiener chaos, extending understanding of collision phenomena.

Findings

Collision measures converge to a non-trivial limit measure

Limit measure characterized via Wiener chaos

Methods inspired by statistical mechanics

Abstract

For an integer $k\ge 2$, let $S^{(1)}, S^{(2)}, \dots, S^{(k)}$ be $k$ independent simple symmetric random walks on $\mathbb{Z}$. A pair $(n,z)$ is called a collision event if there are at least two distinct random walks, namely, $S^{(i)},S^{(j)}$ satisfying $S^{(i)}_n= S^{(j)}_n=z$. We show that under the same scaling as in Donsker's theorem, the sequence of random measures representing these collision events converges to a non-trivial random measure on $[0,1]\times \mathbb{R}$. Moreover, the limit random measure can be characterized using Wiener chaos. The proof is inspired by methods from statistical mechanics, especially, by a partition function that has been developed for the study of directed polymers in random environments.