Directed percolation with incubation times

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.

17 pages, 7 figures. v.2: minor corrections

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Cited by in corpus (8)

• Non-equilibrium Phase Transitions with Long-Range Interactions
• Non-Markovian Levy diffusion in nonhomogeneous media
• Long-range epidemic spreading with immunization
• Nonequilibrium phase transition in a spreading process on a timeline
• Critical dynamics in systems controlled by fractional kinetic equations
• Pattern formation in reaction-diffusion systems in the presence of short-term memory
• Theory of Critical Phenomena with Long-Range Temporal Interaction
• Markovian to non-Markovian phase transition in the operator dynamics of a mobile impurity