Novel non-equilibrium critical behavior in unidirectionally coupled stochastic processes

TL;DR

This paper investigates a hierarchy of unidirectionally coupled directed percolation processes, revealing multicritical behavior with reduced density exponents and employing field-theoretic and simulation methods to analyze critical phenomena.

Contribution

It introduces a new hierarchical model of coupled DP processes, derives mean-field and fluctuation-corrected exponents, and connects these to growth processes and cellular automata.

Findings

Multicritical behavior with density exponents rac{1}{2^{i-1}} at hierarchy level i.

Fluctuation corrections to the second level exponent: rac{1}{2} - rac{\u03b5}{8}.

Monte Carlo simulations confirm scaling exponents in low dimensions.

Abstract

Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension d_c = 4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes A -> A + A, A + A -> A, and A -> \emptyset. We study a hierarchy of such DP processes for particle species A, B,..., unidirectionally coupled via the reactions A -> B, ... (with rates \mu_{AB}, ...). When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents \beta_i which are markedly reduced at each hierarchy level i >= 2. This scenario can be understood on the basis of the mean-field rate equations, which yield \beta_i = 1/2^{i-1} at the multicritical point. We then include fluctuations by using field-theoretic renormalization group techniques in d = 4-\epsilon dimensions. In the active phase, we calculate the fluctuation correction to the density exponent for the second hierarchy level, \beta_2 = 1/2 - \epsilon/8 + O(\epsilon^2). Monte Carlo simulations are then employed to determine the values for the new scaling exponents in dimensions d<= 3, including the critical initial slip exponent. Our theory is connected to certain classes of growth processes and to certain cellular automata, as well as to unidirectionally coupled pair annihilation processes. We also discuss some technical and conceptual problems of the loop expansion and their possible interpretation.