The Field Theory Approach to Percolation Processes

TL;DR

This paper reviews the field theory approach to percolation processes, focusing on epidemic models and their critical behavior near phase transitions, using renormalization group methods to analyze universal properties.

Contribution

It provides a comprehensive overview of constructing field theories for epidemic processes and applying RG techniques to derive critical exponents and scaling functions.

Findings

Critical exponents for directed and isotropic percolation

Universal scaling functions near critical points

Effects of long-range spreading and boundaries

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.