Field Theory of Branching and Annihilating Random Walks

TL;DR

This paper develops a field-theoretic approach to analyze branching and annihilating random walks, revealing different phase behaviors and universality classes depending on the dimension and parity of m.

Contribution

It introduces a systematic field-theoretic and renormalization group framework to study phase transitions in branching-annihilating random walks, including new critical dimensions and universality classes.

Findings

Active phase appears immediately for d>2 with non-trivial crossover exponents.

A non-trivial inactive phase emerges below a critical dimension d_c'~4/3.

Transition to active phase for odd m belongs to directed percolation universality class.

Abstract

We develop a systematic analytic approach to the problem of branching and annihilating random walks, equivalent to the diffusion-limited reaction processes 2A->0 and A->(m+1)A, where m>=1. Starting from the master equation, a field-theoretic representation of the problem is derived, and fluctuation effects are taken into account via diagrammatic and renormalization group methods. For d>2, the mean-field rate equation, which predicts an active phase as soon as the branching process is switched on, applies qualitatively for both even and odd m, but the behavior in lower dimensions is shown to be quite different for these two cases. For even m, and d~2, the active phase still appears immediately, but with non-trivial crossover exponents which we compute in an expansion in eps=2-d, and with logarithmic corrections in d=2. However, there exists a second critical dimension d_c'~4/3 below which a non-trivial inactive phase emerges, with asymptotic behavior characteristic of the pure annihilation process. This is confirmed by an exact calculation in d=1. The subsequent transition to the active phase, which represents a new non-trivial dynamic universality class, is then investigated within a truncated loop expansion. For odd m, we show that the fluctuations of the annihilation process are strong enough to create a non-trivial inactive phase for all d<=2. In this case, the transition to the active phase is in the directed percolation universality class.