Statistics of the one-dimensional Riemann walk

The Riemann walk is the lattice version of the Levy flight. For the one-dim ensional Riemann walk of Levy exponent 0 < alpha <2 we study the statistics of the support, i.e., set of visited sites, after t steps. We consider a w ide 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 --> infinity we obtain the asymptotic power laws for the averages, va riances, and correlations of these observables. Logarithmic correction fact ors appear for alpha =2/3 and alpha =1. Bulk and surface observables have d ifferent power laws for 1 less than or equal to alpha less than or equal to 2. Fluctuations are shown to be universal for 2/3 less than or equal to alp ha less than or equal to2. This means that in the limit t --> infinity the deviations from average DeltaM(t) + M(t) - M(t) are fully described either by a single M independent stochastic process (when 2/3 < alpha less than or equal to1) or by two such processes, one for the bulk and one For the urfa ce observables (when 2/3 < alpha less than or equal to1) or by two such pro cesses, one for the bulk and one for the surface observables (when 1 < alph a <2).