Phase diagram and critical exponents for the pair annihilation model in one and two dimensions

We study the one and two-dimensional pair annihilation model (PAM) with diffusion using the pair approximation (PA) and Monte Carlo simulation. The model is defined on a $d$-dimensional hypercubic lattice where sites can be either occupied by a particle or vacant. The particles are allowed to diffuse with probability $D$; pairs of particles are annihilated with rate $1 $, and particles are created with rate $n\lambda/(2d)$, where $n$ is the number of occupied neighbors. The active and absorbing phases are separated by a continuous phase transition at $\lambda_c(D)$. We obtain the phase diagram and critical exponents for the model, which confirm that PAM belongs to the directed percolation (DP) universality class. The PA for the one and two-dimensional PAM indicates that for a diffusion rate greater than a certain value, $\lambda_c=0$. On the other hand, Katori and Konno [Katori, K., and Konno, N. (1992). Physica A {\bf 186}, 578] showed rigorously that $\lambda_c > 0$ for any diffusion rate, in one dimension. Our simulation results are consistent with this theorem, but suggest that the PA prediction is correct in two dimensions.