The Field Theory Approach to Percolation Processes

Abstract

We review the field theory approach to percolation processes. Specifically, we focus on the so-called simple and general epidemic processes that display continuous non-equilibrium active to absorbing state phase transitions whose asymptotic features are governed respectively by the directed (DP) and dynamic isotropic percolation (dIP) universality classes. We discuss the construction of a field theory representation for these Markovian stochastic processes based on fundamental phenomenological considerations, as well as from a specific microscopic reaction-diffusion model realization. Subsequently we explain how dynamic renormalization group (RG) methods can be applied to obtain the universal properties near the critical point in an expansion about the upper critical dimensions d_c = 4 (DP) and 6 (dIP). We provide a detailed overview of results for critical exponents, scaling functions, crossover phenomena, finite-size scaling, and also briefly comment on the influence of long-range spreading, the presence of a boundary, multispecies generalizations, coupling of the order parameter to other conserved modes, and quenched disorder.

54 pages, figures included

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• Crossover from Isotropic to Directed Percolation
• Equation of state for directed percolation
• Universal scaling behavior of directed percolation around the upper critical dimension
• Field-Theoretic Techniques in the Study of Critical Phenomena
• Logarithmic Corrections in Directed Percolation
• Random Resistor-Diode Networks and the Crossover from Isotropic to Directed Percolation
• Crossover from Self--Similar to Self--Affine Structures in Percolation

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