Number of distinct and common sites visited by $N$ independent random walkers

TL;DR

This paper analyzes the behavior of the number of distinct and common sites visited by N independent random walkers in various dimensions, revealing phase transitions and exact distributions, with extensions to related stochastic processes.

Contribution

It provides exact calculations of mean numbers of visited sites and uncovers phase transitions in the growth behavior of common sites visited by the walkers.

Findings

Exact mean values of visited sites for large t.

Identification of a phase transition in the (N, d) plane.

Full distribution of visited sites in 1D computed.

Abstract

In this Chapter, we consider a model of $N$ independent random walkers, each of duration $t$, and each starting from the origin, on a lattice in $d$ dimensions. We focus on two observables, namely $D_N(t)$ and $C_N(t)$ denoting respectively the number of distinct and common sites visited by the walkers. For large $t$, where the lattice random walkers converge to independent Brownian motions, we compute exactly the mean $\langle D_N(t) \rangle$ and $\langle C_N(t) \rangle$. Our main interest is on the $N$-dependence of these quantities. While for $\langle D_N(t) \rangle$ the $N$-dependence only appears in the prefactor of the power-law growth with time, a more interesting behavior emerges for $\langle C_N(t) \rangle$. For this latter case, we show that there is a ``phase transition'' in the $(N, d)$ plane where the two critical line $d=2$ and $d=d_c(N) = 2N/(N-1)$ separate three phases of the growth of $\langle C_N(t)\rangle$. The results are extended to the mean number of sites visited exactly by $K$ of the $N$ walkers. Furthermore in $d=1$, the full distribution of $D_N(t)$ and $C_N(t)$ are computed, exploiting a mapping to the extreme value statistics. Extensions to two other models, namely $N$ independent Brownian bridges and $N$ independent resetting Brownian motions/bridges are also discussed.