Field theory of directed percolation with long-range spreading

Abstract

It is well established that the phase transition between survival and extinction in spreading models with short-range interactions is generically associated with the directed percolation (DP) universality class. In many realistic spreading processes, however, interactions are long ranged and well described by Lévy-flights, i.e., by a probability distribution that decays in $d$ dimensions with distance $r$ as $r^{-d-\sigma }$. We employ the powerful methods of renormalized field theory to study DP with such long range, Lévy-flight spreading in some depth. Our results unambiguously corroborate earlier findings that there are four renormalization group fixed points corresponding to, respectively, short-range Gaussian, Lévy Gaussian, short-range DP and Lévy DP, and that there are four lines in the $(\sigma ,d)$ plane which separate the stability regions of these fixed points. When the stability line between short-range DP and Lévy DP is crossed, all critical exponents change continuously. We calculate the exponents describing Lévy DP to second order in $\epsilon$-expansion, and we compare our analytical results to the results of existing numerical simulations. Furthermore, we calculate the leading logarithmic corrections for several dynamical observables.

12 pages, 3 figures

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