Directed percolation with incubation times

TL;DR

This paper develops a field theory for directed percolation with non-Markovian incubation times modeled by Levy distributions, revealing a new universality class with continuously varying critical exponents.

Contribution

It introduces a non-Markovian extension of directed percolation, formulates a corresponding field theory, and analyzes its critical behavior and scaling relations.

Findings

Critical exponents vary continuously with Levy parameter.

Absence of field renormalization at one-loop order.

Derivation of a new scaling relation for the critical exponents.

Abstract

We introduce a model for directed percolation with a long-range temporal diffusion, while the spatial diffusion is kept short ranged. In an interpretation of directed percolation as an epidemic process, this non-Markovian modification can be understood as incubation times, which are distributed accordingly to a Levy distribution. We argue that the best approach to find the effective action for this problem is through a generalization of the Cardy-Sugar method, adding the non-Markovian features into the geometrical properties of the lattice. We formulate a field theory for this problem and renormalize it up to one loop in a perturbative expansion. We solve the various technical difficulties that the integrations possess by means of an asymptotic analysis of the divergences. We show the absence of field renormalization at one-loop order, and we argue that this would be the case to all orders in perturbation theory. Consequently, in addition to the characteristic scaling relations of directed percolation, we find a scaling relation valid for the critical exponents of this theory. In this universality class, the critical exponents vary continuously with the Levy parameter.