considerations hold for all other sites. Accounting for all of these, we write $$\begin{aligned}
\frac{dp_{i,j}}{dt}=\frac{\gamma(-2J)}{2}(-12 p_{i,j}+p_{i+2,j}+p_{i-2,j}+p_{i,j+2}+p_{i,j-2}  \nonumber  \\
+2p_{i+1,j+1}+2p_{i+1,j-1}+2p_{i-1,j+1}+2p_{i-1,j-1}), \nonumber\end{aligned}$$ which, in the continuum limit, yields $D=4\gamma(-2J)$. We can expect that the properties of the memory improve by lowering the value of $\gamma(0)$, but only as long as $\gamma(0)\gtrsim 4\gamma(-2J)$. In the interacting case, $J$ can be replaced by an appropriate excitation energy (e.g., a mean-field gap, see Sec. \[sec:mf\]).

Lifetime of the non-interacting model {#sec:lifetime_formula}
-------------------------------------

We can now express the error creation rate in terms of the diffusion constant. An isolated anyon can have either one or three $\sigma_x$-errors at its plaquette spins. In the first case, the anyon can hop to a neighboring site either by creating an error on one of the initially unaffected spins, or by removing the one pre-existing error. Therefore, such an anyon contributes to the error rate with $2D = 3D-D$. If three $\sigma_x$-errors are present, an opposite rate $-2D$ is obtained from an analogous reasoning. However, three-error plaquettes can be expected to be less likely: they require that an anyon hopped on a plaquette with two pre-existing errors from one of the two directions without errors. From the above discussion, it 