On the number of common sites visited by N random walkers

TL;DR

This paper explores the asymptotic behavior of the mean number of common sites visited by N independent random walkers in a d-dimensional lattice, using fractal intersection theory to analyze different regimes.

Contribution

It introduces a fractal intersection approach to derive asymptotic power laws for the number of common sites visited by multiple random walkers.

Findings

Different regimes of common site visitation are characterized.

Asymptotic power laws are derived for various regimes.

The approach unifies previous results using fractal intersection concepts.

Abstract

Majumdar and Tamm [Phys. Rev. E 86 021135 (2012), arXiv:1206.6184] recently obtained analytical expressions for the mean number of common sites W_N(t) visited up to time t by N independent random walkers starting from the origin of a d-dimensional lattice. In this short note I show how the different regimes and the corresponding asymptotic power laws can be retrieved using the notion of fractal intersection.