On the moments of the meeting time of independent random walks in random environment

TL;DR

This paper studies the distribution and moments of the first meeting time of multiple independent random walks in a Sinai regime environment, revealing a phase transition in the finiteness of moments based on the number of walks.

Contribution

It provides a detailed analysis of the tail behavior and moments of the meeting time, including convergence results and explicit moment thresholds, in a complex random environment.

Findings

Tail distribution converges to a functional of Brownian motion.

Explicit law of the limiting functional is derived.

Moments of the meeting time are finite or infinite depending on the exponent and number of walks.

Abstract

We consider, in the continuous time version, $\gamma$ independent random walks on $\mathbb{Z_+}$ in random environment in the Sinai's regime. Let $T_\gam$ be the first meeting time of one pair of the $\gamma$ random walks starting at different positions. We first show that the tail of the quenched distribution of $T_\gamma$, after a suitable rescaling, converges in probability, to some functional of the Brownian motion. Then we compute the law of this functional. Eventually, we obtain results about the moments of this meeting time. Being $\Eo$ the quenched expectation, we show that, for almost all environments $\omega$, $\Eo[T_\gamma^{c}]$ is finite for $c<\gamma(\gamma-1)/2$ and infinite for $c>\gamma(\gamma-1)/2$.