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, 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 an 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.