Renormalized field theory is a most effective framework to carry out asymptotic analysis of non-equilibrium nearly critical systems, especially in high orders of perturbation theory. Here, we review some subtle, slippery and non-conventional aspects of this approach. We present construction of the field-theoretic representation of certain Langevin-type stochastic equations with additive and multiplicative random sources as well as master equations of various birth–death processes. Application of the field-theoretic renormalization group combined with the short-distance operator-product expansion to the analysis of asymptotic scaling behavior is reviewed for passive scalar fields advected by various velocity ensembles, including Kraichnan’s rapid-change model and the stochastic Navier–Stokes equation. Infinite sets of anomalous exponents were calculated within regular expansions up to third order. Effects of anisotropy, finite correlation time and compressibility are discussed. The representation of the Kolmogorov constant and the skewness factor suitable for perturbative renormalization-group calculation and the second-order results are presented in a reasonable agreement with experiments in fully developed hydrodynamic turbulence. The recent third-order results for the critical exponents for the directed percolation process are presented; paradigmatic models for irreversible reaction–diffusion processes are discussed with the account of advection in various random velocity fields.

Peer reviewed