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

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.

26 pages, 6 figures, 45 references

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Cited by in corpus (4)

• Record statistics of a strongly correlated time series: random walks and Lévy flights
• Exact distributions of cover times for $N$ independent random walkers in one dimension
• Reaction-diffusion on the fully-connected lattice: $A+A\to A$
• Two-species diffusion-annihilation process on the fully-connected lattice: probability distributions and extreme value statistics