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<α<2 we study the statistics of the support, i.e., 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 adjacent visited sites. For t→∞ we obtain the asymptotic power laws for the averages, variances, and correlations of these observables. Logarithmic correction factors appear for α=2/3 and α=1. Bulk and surface observables have different power laws for 1≤α<2. Fluctuations are shown to be universal for 2/3 ≤α<2. This means that in the limit t→∞ the deviations from average ΔM(t)≡M(t)−M-0304;(t-0304;) are fully described either by a single M independent stochastic process (when 2/3 <α≤1) or by two such processes, one for the bulk and one for the surface observables (when 1<α<2).