Two-Particle Dispersion in Model Velocity Fields

We consider two-particle dispersion in a velocity field, where the relative two-point velocity scales according to $v^{2}(r)\propto r^{\alpha}$ and the corresponding correlation time scales as $\tau (r)\propto r^{\beta}$, and fix $\alpha =2/3$, as typical for turbulent flows. We show that two generic types of dispersion behavior arize: For $\alpha /2+\beta < 1$ the correlations in relative velocities decouple and the diffusion approximation holds. In the opposite case, $\alpha /2+\beta >1$, the relative motion is strongly correlated. The case of Kolmogorov flows corresponds to a marginal, nongeneric situation.Comment: 4 pages, 4 figures, Late

Canonical fitness model for simple scale-free graphs

We consider a fitness model assumed to generate simple graphs with power-law heavy-tailed degree sequence: P(k) \propto k^{-1-\alpha} with 0 < \alpha < 1, in which the corresponding distributions do not posses a mean. We discuss the situations in which the model is used to produce a multigraph and examine what happens if the multiple edges are merged to a single one and thus a simple graph is built. We give the relation between the (normalized) fitness parameter r and the expected degree \nu of a node and show analytically that it possesses non-trivial intermediate and final asymptotic behaviors. We show that the model produces P(k) \propto k^{-2} for large values of k independent of \alpha. Our analytical findings are confirmed by numerical simulations.Comment: 6 pages, 2 figures; published in Phys. Rev. E. To improve readability, formulas and text were added between Eq. (1) and (2

L\'{e}vy flights as subordination process: first passage times

We obtain the first passage time density for a L\'{e}vy flight random process from a subordination scheme. By this method, we infer the asymptotic behavior directly from the Brownian solution and the Sparre Andersen theorem, avoiding explicit reference to the fractional diffusion equation. Our results corroborate recent findings for Markovian L\'{e}vy flights and generalize to broad waiting times.Comment: 4 pages, RevTe