is justified to estimate the rate at which errors are created to be of order $D$ per anyon.

Finally, assuming $N$ diffusing anyons present in the system, the fraction $f$ of spins affected by a $\sigma_{x}$-error after a time $t$ is estimated as $f \simeq N D t/2 L^2$ and the error correction fails when $f$ is larger than some critical value $f_c$ [@Dennis2002]. This gives a lifetime $\tau$ for the memory $$\label{T_estimate}
\tau \simeq 2 f_{c} \frac{e^{\beta J}+1}{\max\{ \gamma(0),4\gamma(-2J)\}},$$ where we have replaced the factor $N/L^2$ by the equilibrium occupation $\langle n_p \rangle = 1/(e^{\beta J}+1)$.

An analogous result can be obtained based on the following different reasoning [@Alicki2009; @Hamma2009]. The distance between the two anyons of a pair after a time $\tau$ is of order $\Delta\ell=\sqrt{D \tau}$ and is required to be much smaller than the average anyon separation $\sim \sqrt{L^2/N}$. This gives $\tau \ll (e^{\beta J}+1)/\max \{ \gamma(0),4\gamma(-2J)\}$. Interestingly, this upper bound coincides with the right-hand side of Eq. (\[T\_estimate\]) if the probability for each spin to be flipped is $\frac12$ (which is realized at long times).

Equation (\[T\_estimate\]) generally gives reasonable estimates of the memory lifetime. For example, the value $f_c \simeq 0.11$ of [@Dennis2002] yields 