A multivariate extension of the Erd\"os-Taylor theorem

TL;DR

This paper extends the Erd"os-Taylor theorem to multiple independent random walks, showing that their pairwise collision times, when properly scaled, converge jointly to independent exponential distributions, revealing a multivariate structure.

Contribution

It introduces a multivariate extension of the Erd"os-Taylor theorem, demonstrating joint convergence of scaled collision local times for multiple random walks.

Findings

Joint convergence of scaled pairwise collision times to independent exponentials

Multivariate distributional limit for collision local times

Connections established to directed polymers in random environments

Abstract

The Erd\"os-Taylor theorem [Acta Math. Acad. Sci. Hungar, 1960] states that if $\mathsf{L}_N$ is the local time at zero, up to time $2N$, of a two-dimensional simple, symmetric random walk, then $\tfrac{\pi}{\log N} \,\mathsf{L}_N$ converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if $\mathsf{L}_N^{(1,2)}=\sum_{n=1}^N 1_{\{S_n^{(1)}= S_n^{(2)}\}}$, then $\tfrac{\pi}{\log N}\, \mathsf{L}^{(1,2)}_N$ converges in distribution to an exponential random variable of parameter one. We prove that for every $h \geq 3$, the family $ \big\{ \frac{\pi}{\log N} \,\mathsf{L}_N^{(i,j)} \big\}_{1\leq i<j\leq h}$, of logarithmically rescaled, two-body collision local times between $h$ independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erd\"os-Taylor theorem. We also discuss connections to directed polymers in random environments.