Full distribution of the number of distinct sites visited by a random walker in dimension $d \ge 2$

TL;DR

This paper analyzes the full distribution of the number of distinct sites visited by a random walk in dimensions two and higher, revealing different large deviation behaviors in the tails and validating predictions with numerical simulations.

Contribution

It provides theoretical predictions for the large deviation principles governing the distribution of visited sites, including anomalous scaling in the left tail for dimensions greater than two.

Findings

Right tail follows standard large deviation principle with exponential decay.

Left tail exhibits anomalous scaling with a different exponential decay.

Analytical results obtained for high dimensions and small visited site fractions.

Abstract

We study the full distribution $P_M(S)$ of the number of distinct sites $S$ visited by a random walker on a $d$-dimensional lattice after $M$ steps. We focus on the case $d \ge 2$, and we are interested in the long-time limit $M \gg 1$. Our primary interest is the behavior of the right and left tails of $P_M(S)$, corresponding to $S$ larger and smaller than its mean value, respectively. We present theoretical arguments that predict that in the right tail, a standard large-deviation principle (LDP) $P_{M}\left(S\right)\sim e^{-M\Phi\left(S/M\right)}$ is satisfied (at $M \gg 1$) for $d\ge2$, while in the left tail, the scaling behavior is $P_{M}\left(S\right)\sim e^{-M^{1-2/d}\Psi\left(S/M\right)}$, corresponding to a LDP with anomalous scaling, for $d>2$. We also obtain bounds for the scaling functions $\Phi(a)$ and $\Psi(a)$, and obtain analytical results for $\Phi(a)$ in the high-dimensional limit $d \gg 1$, and for $\Psi(a)$ in the limit $a \ll 1$ (describing the far left tail). Our predictions are validated by numerical simulations using importance sampling algorithms.