Branching-rate expansion around annihilating random walks

We present some exact results for branching and annihilating random walks. We compute the nonuniversal threshold value of the annihilation rate for having a phase transition in the simplest reaction-diffusion system belonging to the directed percolation universality class. Also, we show that the accepted scenario for the appearance of a phase transition in the parity conserving universality class must be improved. In order to obtain these results we perform an expansion in the branching rate around pure annihilation, a theory without branching. This expansion is possible because we manage to solve pure annihilation exactly in any dimension.

Figure 1

Generic form of a diagram for ${\Gamma }^{(n,m)}$ including at least one loop. (Left) Diagrammatic representation of a generic ${\Gamma }^{(n,m)}$ vertex. (Right) General structure for ${\Gamma }^{(n,m)}$ in PA; the black blob is a connected and amputated Green function that has to comply with some requisites (see text).

Figure 2

Closed equation for ${\Gamma }^{(1,3)}$ in BARW-PC.

Figure 3

Results for $d_{\sigma }$, showing there is no change in the RG relevance for the branching rate $\sigma$.

Figure 4

Closed equation for ${\Gamma }^{(1,1)}$ in BARW-DP at $O\left(\sigma \right)$.