A systematic theory for the diffusion-limited reaction processes $A+A\to 0$ and $A\to (m+1\mathrm{})A$ 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 $d_c=2\mathrm{}$ dimensions; but below $d_c^{\prime }\approx 4/3$ a nontrivial transition to an inactive phase governed by power law behavior appears. For odd $m$, there is a dynamic phase transition for any $d\le 2$ which is described by the directed percolation universality class.

• Received 6 August 1996

DOI:https://doi.org/10.1103/PhysRevLett.77.4780