# Groupe de Physique Statistique

## Equipe 106, Institut Jean Lamour

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### Articles dans des revues à comité de lecture

 Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice Turban L. J. Phys. A: Math. Theor. 47 (2014) 385004 DOI : 10.1088/1751-8113/47/38/385004 ArXiv : arxiv:1409.3718 [PDF] 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 ($tto infty$, $Nto infty$ 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 ${{d}_{{rm c}}}=2$.