Records for the number of distinct sites visited by a random walk on the fully-connected lattice

TL;DR

This paper analyzes the statistical properties of the number of distinct sites visited by a random walk on a fully-connected lattice, deriving probability distributions and their asymptotic behaviors for various covering scenarios.

Contribution

It introduces exact generating functions for record statistics in a random walk on a fully-connected lattice, revealing Gaussian and Gumbel distributions in different regimes.

Findings

Distributions for the number of visited sites and record values are derived using generating functions.

At large scales, the distributions converge to Gaussian densities.

Record times follow Gumbel distributions at total covering and crossover to Gaussian at partial covering.

Abstract

We consider a random walk on the fully-connected lattice with $N$ sites and study the time evolution of the number of distinct sites $s$ visited by the walker on a subset with $n$ sites. A record value $v$ is obtained for $s$ at a record time $t$ when the walker visits a site of the subset for the first time. The record time $t$ is a partial covering time when $v<n$ and a total covering time when $v=n$. The probability distributions for the number of records $s$, the record value $v$ and the record (covering) time $t$, involving $r$-Stirling numbers, are obtained using generating function techniques. The mean values, variances and skewnesses are deduced from the generating functions. In the scaling limit the probability distributions for $s$ and $v$ lead to the same Gaussian density. The fluctuations of the record time $t$ are also Gaussian at partial covering, when $n-v={\mathrm O}(n)$. They are distributed according to the type-I Gumbel extreme-value distribution at total covering, when $v=n$. A discrete sequence of generalized Gumbel distributions, indexed by $n-v$, is obtained at almost total covering, when $n-v={\mathrm O}(1)$. These generalized Gumbel distributions are crossing over to the Gaussian distribution when $n-v$ increases.