Number of Common Sites Visited by N Random Walkers

TL;DR

This paper analytically investigates the asymptotic behavior of the mean number of common sites visited by N independent random walkers in various dimensions, revealing three distinct regimes separated by critical lines and confirmed through simulations.

Contribution

It provides a comprehensive analytical characterization of the growth regimes of common visited sites by multiple random walkers across different dimensions, including critical behaviors.

Findings

Identified three regimes of growth for W_N(t) in (N-d) plane.

Derived explicit asymptotic forms for each regime.

Validated analytical results with numerical simulations.

Abstract

We compute analytically the mean number of common sites, W_N(t), visited by N independent random walkers each of length t and all starting at the origin at t=0 in d dimensions. We show that in the (N-d) plane, there are three distinct regimes for the asymptotic large t growth of W_N(t). These three regimes are separated by two critical lines d=2 and d=d_c(N)=2N/(N-1) in the (N-d) plane. For d<2, W_N(t)\sim t^{d/2} for large t (the N dependence is only in the prefactor). For 2<d<d_c(N), W_N(t)\sim t^{\nu} where the exponent \nu= N-d(N-1)/2 varies with N and d. For d>d_c(N), W_N(t) approaches a constant as t\to \infty. Exactly at the critical dimensions there are logaritmic corrections: for d=2, we get W_N(t)\sim t/[\ln t]^N, while for d=d_c(N), W_N(t)\sim \ln t for large t. Our analytical predictions are verified in numerical simulations.