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A systematic theory for the diffusion-limited reaction processes A+A→0 and A→(m+1)A is developed. Fluctuations are taken into account via the field-theoretic dynamical renormalization group. For even m, the mean field rate equation, which predicts only an active phase, remains qualitatively correct near dc0ex0ex=0ex0ex2 dimensions; but below dc^'≈4/3 a nontrivial transition to an inactive phase governed by power law behavior appears. For odd m, there is a dynamic phase transition for any d≤2 which is described by the directed percolation universality class.
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Cardy et al. (1996) studied this question.
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