How often can two independent elephant random walks on $\mathbb{Z}$ meet?

TL;DR

This paper investigates the meeting frequency of two independent elephant random walks on the integer lattice, revealing a phase transition at the memory parameter $p=3/4$ and providing asymptotic behavior of their distance.

Contribution

It establishes a clear criterion based on the memory parameter for whether the walks meet finitely or infinitely often, and derives asymptotic results for their separation.

Findings

Finitely or infinitely often meeting depends on whether p > 3/4.

Asymptotic behavior of the distance between the walks is characterized.

Phase transition at p=3/4 for the recurrence of meetings.

Abstract

We show that two independent elephant random walks on the integer lattice $\mathbb{Z}$ meet each other finitely often or infinitely often depends on whether the memory parameter $p$ is strictly larger than $3/4$ or not. Asymptotic results for the distance between them are also obtained.