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Phase diagram and critical behavior of the pair annihilation model

We study the critical behavior of the pair annihilation model (PAM) with diffusion in one, two and three dimensions, using the pair approximation (PA) and Monte Carlo simulation. Of principal interest is the dependence of the critical creation rate, lambda_c, on the diffusion probability D, in particular, whether survival is possible at arbitrarily small creation rates, for sufficiently rapid diffusion. Whilst the PA predicts that in any spatial dimension d \geq 1, lambda_c -> 0 at some diffusion probability D* < 1, Katori and Konno [Physica A {\bf 186}, 578 (1992)] showed rigorously that for d \leq 2, one has lambda_c > 0 for any D<1. Our simulation results are consistent with this theorem. In two dimensions, the extinction region becomes narrow as D approaches unity, following lambda_c \propto \exp[- \rm{const.}/(1-D)^γ], with gamma = 1.41(2). In three dimensions we find D* = 0.333(3). The simulation results confirm that the PAM belongs to the directed percolation universality class.