Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice

TL;DR

This paper derives the probability distribution of the number of distinct sites visited by a random walk on a finite fully-connected lattice, revealing Gaussian fluctuations in the scaling limit.

Contribution

It provides an exact solution for the joint distribution of visited sites and introduces finite-size scaling analysis for the problem.

Findings

Joint distribution of visited sites and single-visit sites is Gaussian in the scaling limit.

Finite-size effects follow a bivariate Gaussian distribution.

Results are expected to extend to higher-dimensional periodic lattices.

Abstract

The probability distribution of the number $s$ of distinct sites visited up to time $t$ by a random walk on the fully-connected lattice with $N$ sites is first obtained by solving the eigenvalue problem associated with the discrete master equation. Then, using generating function techniques, we compute the joint probability distribution of $s$ and $r$, where $r$ is the number of sites visited only once up to time $t$. Mean values, variances and covariance are deduced from the generating functions and their finite-size-scaling behaviour is studied. Introducing properly centered and scaled variables $u$ and $v$ for $r$ and $s$ and working in the scaling limit ($t\to\infty$, $N\to\infty$ with $w=t/N$ fixed) the joint probability density of $u$ and $v$ is shown to be a bivariate Gaussian density. It follows that the fluctuations of $r$ and $s$ around their mean values in a finite-size system are Gaussian in the scaling limit. The same type of finite-size scaling is expected to hold on periodic lattices above the critical dimension $d_{\rm c}=2$.