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Published on: December 16, 2022
Directed percolation with incubation times
1Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK. andrea@mpipks-dresden.mpg.de
Insights
We present a new model for directed percolation with long-range temporal diffusion, incorporating incubation times using Lévy distributions. This model shows critical exponents that continuously vary with the Lévy parameter.
Area of Science:
- Statistical Physics
- Complex Systems
- Epidemic Modeling
Background:
- Directed percolation models critical phenomena in systems with quenched disorder.
- Standard models assume short-range interactions and Markovian processes.
- Epidemic processes often exhibit non-Markovian features like incubation periods.
Purpose of the Study:
- Introduce a novel directed percolation model with long-range temporal diffusion.
- Incorporate non-Markovian dynamics, specifically Lévy-distributed incubation times.
- Develop a field theory and renormalization group approach for this modified system.
Main Methods:
- Generalization of the Cardy-Sugar method to include non-Markovian temporal diffusion.
- Formulation of a field theory for the modified directed percolation model.
- One-loop perturbative renormalization group analysis, employing asymptotic analysis for divergences.
Main Results:
- Demonstrated the absence of field renormalization at one-loop and argued for its absence at all orders.
- Derived characteristic scaling relations for directed percolation.
- Identified a new scaling relation for critical exponents, showing continuous variation with the Lévy parameter.
Conclusions:
- The proposed model offers a framework for studying critical phenomena with non-Markovian temporal effects.
- The continuous variation of critical exponents with the Lévy parameter introduces a new universality class.
- This work provides insights into epidemic processes with realistic incubation time distributions.
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 Lévy 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 Lévy parameter.
