in all cases.

![[]{data-label="lm1d"}](lm1d.eps){width="0.8\hsize"}

![[]{data-label="11site"}](11site.eps){width="0.8\hsize"}

![[]{data-label="lm2d"}](lambda-mu.eps){width="0.8\hsize"}

The Diffusive SCP
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Although the one-site MFT predicts a discontinuous phase transition in the 2SCP in any number of dimensions, such a transition is not possible in one-dimensional systems with short-range interactions and free of boundary fields [@hinrichsen]. In one dimension the active-absorbing transition should be continuous, as we have indeed verified in simulations. In two dimensions ($d=2$), previous studies did not reveal any evidence for a discontinuous transition. These studies did not, however, include diffusion, which is expected to facilitate the appearance of discontinuous transitions. Here we study the 2SCP with diffusion on the square lattice.

We modify the process so that, in addition to creation and death, each individual can hop to one of its NN sites at rate $D$. In the simulation algorithm for the diffusive 2SCP, we maintain two lists, one of singly and another of doubly occupied sites. Let $N_s$ and $N_d$ denote, respectively, the numbers of such sites, so that $N_p = N_s + 2 N_d$ is the total number of individuals. The total rate of (attempted) transitions is $\lambda N_p + N_s + 2\mu N_d + D N_p\equiv 1/\Delta t$, where $\Delta t$ is the time 