FIELD-THEORY OF BRANCHING AND ANNIHILATING RANDOM-WALKS

We develop a systematic analytic approach to the problem of branching and annihilating random walks, equivalent to the diffusion-limited rea ction processes 2A --> circle divide and A --> (m + 1) A, where m grea ter than or equal to 1. Starting from the master equation, a field-the oretic representation of the problem is derived, and fluctuation effec ts are taken into account via diagrammatic and renormalization group m ethods. For d > 2, the mean-field rate equation, which predicts an act ive phase as soon as the branching process is switched on, applies qua litatively for both even and odd m, but the behavior in lower dimensio ns 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 nontri vial crossover exponents which we compute in an expansion in epsilon = 2 - d and with logarithmic corrections in d = 2. However, there exist s a second critical dimension d(c)' approximate to 4/3 below which a n ontrivial inactive phase emerges, with asymptotic behavior characteris tic of the pure annihilation process. This is confirmed by an exact ca lculation in d = 1. The subsequent transition to the active phase, whi ch represents a new nontrivial dynamic universality class, is then inv estigated within a truncated loop expansion, which appears to give st correct qualitative picture. The model with m = 2 is also generalized to N species of particles, which provides yet another universality cla ss and which is exactly solvable in the limit N --> infinity. For odd m, we show that the fluctuations of the annihilation process are stron g enough to create a nontrivial inactive phase for all d less than or equal to 2. In this case, the transition to the active phase is in the directed percolation universality class. Finally, we study the modifi cation when the annihilation reaction is 3A --> circle divide. When m = 0 (mod 3) the system is always in its active phase, but with logarit hmic crossover corrections for d = 1, while the other cases should exh ibit a directed percolation transition out of a fluctuation-driven ina ctive phase.