A Generalized Epidemic Process and Tricritical Dynamic Percolation

TL;DR

This paper introduces a generalized epidemic process with a weakened state, leading to a tricritical point that separates continuous and discontinuous percolation transitions, and provides a detailed field-theoretic analysis of this universality class.

Contribution

It extends the epidemic process model by adding a weakened state, revealing a tricritical point with a new universality class and calculating its critical exponents using renormalized field theory.

Findings

Identification of a tricritical point in the generalized epidemic process.

Calculation of critical exponents and logarithmic corrections at the upper critical dimension.

Demonstration that order parameter exponents differ at the tricritical point.

Abstract

The renowned general epidemic process describes the stochastic evolution of a population of individuals which are either susceptible, infected or dead. A second order phase transition belonging to the universality class of dynamic isotropic percolation lies between endemic or pandemic behavior of the process. We generalize the general epidemic process by introducing a fourth kind of individuals, viz. individuals which are weakened by the process but not yet infected. This sensibilization gives rise to a mechanism that introduces a global instability in the spreading of the process and therefore opens the possibility of a discontinuous transition in addition to the usual continuous percolation transition. The tricritical point separating the lines of first and second order transitions constitutes a new universality class, namely the universality class of tricritical dynamic isotropic percolation. Using renormalized field theory we work out a detailed scaling description of this universality class. We calculate the scaling exponents in an $\epsilon$-expansion below the upper critical dimension $d_{c}=5$ for various observables describing tricritical percolation clusters and their spreading properties. In a remarkable contrast to the usual percolation transition, the exponents $\beta$ and ${\beta}^{\prime}$ governing the two order parameters, viz. the mean density and the percolation probability, turn out to be different at the tricritical point. In addition to the scaling exponents we calculate for all our static and dynamic observables logarithmic corrections to the mean-field scaling behavior at $d_c=5$.