Statistics of the one-dimensional Riemann walk

Details Statistics of the one-dimensional Riemann walk Statistics of the one-dimensional Riemann walk

2000-03-09

arxiv.org/abs/cond-mat/0003153v1

J. Stat. Phys. 102 (2001) 259

Physics

Condensed Matter

Statistical Mechanics

23 pages, no figures

Scientific paper

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).

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