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

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 $d_{\mathrm{c}}=2$.

20 pages, 5 figures

References in corpus (3)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Exact distributions of the number of distinct and common sites visited by N independent random walkers
• Overlap of two Brownian trajectories: exact results for scaling functions

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• Exact distributions of cover times for $N$ independent random walkers in one dimension
• Records for the number of distinct sites visited by a random walk on the fully-connected lattice
• Reaction-diffusion on the fully-connected lattice: $A+A\to A$