Field theory of directed percolation with long-range spreading

TL;DR

This paper develops a field-theoretic framework for understanding directed percolation with long-range Levy-flight interactions, identifying phase transition regimes and calculating critical exponents with analytical methods.

Contribution

It extends the directed percolation universality class to include long-range Levy-flight spreading using renormalized field theory, identifying fixed points and stability regions.

Findings

Four renormalization group fixed points identified.

Critical exponents vary continuously across stability lines.

Analytical exponents match numerical simulations.

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 L\'{e}vy-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, L\'{e}vy-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, L\'{e}vy Gaussian, short-range DP and L\'{e}vy 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 L\'{e}vy DP is crossed, all critical exponents change continuously. We calculate the exponents describing L\'{e}vy 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.