Statistics of the one-dimensional Riemann walk
Abstract
The Riemann walk is the lattice version of the Levy flight. For the one-dimensional Riemann walk of Levy exponent 0<\alpha<2 we study the statistics of the support, i.e. the set of visited sites, after t steps. We consider a wide class of support related observables M(t), including the number S(t) of visited sites and the number I(t) of sequences of visited sites. For t->\infty we obtain the asymptotic power laws for the averages, variances, and correlations of these observables. Logarithmic correction factors appear for \alpha=2/3 and \alpha=1. Bulk and surface observables have different power laws for 1\leq\alpha<2. Fluctuations are shown to be universal for 2/3\leq\alpha<2. This means that in the limit t->\infty the deviations from average \DeltaM(t) are fully described (i) either by a single M independent stochastic process (when 2/3\leq\alpha\leq 1) (ii) or by two such processes, one for the bulk and one for the surface observables (when 1<\alpha<2).
Cite
@article{arxiv.cond-mat/0003153,
title = {Statistics of the one-dimensional Riemann walk},
author = {A. M. Mariz and F. van Wijland and H. J. Hilhorst and S. R. Gomes Junior and C. Tsallis},
journal= {arXiv preprint arXiv:cond-mat/0003153},
year = {2010}
}
Comments
23 pages, no figures