Field theory of branching and annihilating random walks

Abstract

We develop a systematic analytic approach to the problem of branching and annihilating random walks, equivalent to the diffusion–limited reaction processes 2 A → ∅ 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 near 2, the active phase still appears immediately, but with non–trivial crossover exponents which we compute in an expansion in ϵ = 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, which appears to give a correct qualitative picture. The model with m = 2 is also generalized to N species of particles, which provides yet another universality class and which is exactly solvable in the limit N → ∞. 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. Finally, we study the modification when the annihilation reaction is 3 A → ∅. When m = 0 (mod 3) the system is always in its active phase, but with logarithmic crossover corrections for d = 1, while the other cases should exhibit an directed percolation transition out of a fluctuation–driven inactive phase.