on this range it yields $\lambda_c$ independent of $\mu$, contrary to simulations, which show $\lambda_c$ varying smoothly with $\mu$. For $n=6$ the transition is predicted to be continuous for $\mu < 0.45$, [*discontinuous*]{} for $0.45\leq\mu < 0.88 $, and again continuous for $0.88 \leq \mu \leq 1$. (Note that on the latter interval $\lambda_c$ is again independent of $\mu$). Thus the $n=6$ approximation exhibits the same qualitative problems as for $n=2$, despite the overall improvement. The four-site approximation on the square lattice, shown in Fig. \[lm2d\], furnishes a reasonable prediction for the phase boundary, but suffers from similar defects: for $\mu < 0.66$ the transition is discontinuous, while for $\mu \geq 0.7$, $\lambda_c$ is independent of $\mu$.

In the context of absorbing-state phase transitions, we generally look to MFT as a guide to the overall phase diagram, expecting the critical point to have the correct order of magnitude and, perhaps more importantly, the nature (continuous or discontinuous) of the transition to be predicted correctly. The latter criterion is not always satisfied, however [@trpcr2009]. In light of this, and in the hope of devising a more reliable approximation method that is still relatively simple to apply, we consider analyses based on 