The field theory approach to percolation processes

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Date

2005-01-01

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Academic Press – Elsevier

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 nonequilibrium 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 dc = 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.

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Keywords

Physics, Multidisciplinary, Physics, percolation, epidemic processes, directed percolation, dynamic isotropic percolation, active to absorbing phase transitions, renormalization group theory, dynamic critical phenomena, crossover, STOCHASTIC-EVOLUTION PROCESSES, UNIVERSAL SCALING BEHAVIOR, PAIR-CONTACT PROCESS, ORDER EPSILON-TERMS, DIRECTED PERCOLATION, RENORMALIZATION-GROUP, PHASE-TRANSITIONS, DYNAMICAL PERCOLATION, CELLULAR AUTOMATA, ABSORBING STATES

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