on $T$.](criticalfraction.pdf){width="0.95\columnwidth"}

Also in the dynamical simulations of the non-interacting model (see Fig. \[nonint\_fig\]), we observe a sharp transition in time similar to Fig. \[Z\_vs\_f\]. Starting from an initial state without errors, the thermal environment introduces a growing number of spin-flips which eventually cause the memory to fail. This occurs again at a certain threshold probability $f_c$ which is for this case, however, different from $0.1$, see the inset of Fig. \[Z\_vs\_f\]. To understand this difference, we note that a main mechanism for the creation of errors is the diffusion of anyons. Clearly, errors created by the anyons in their diffusive motion have strong spatial correlations, rather than being independent and uniformly distributed across the memory. We find that such correlations yield values of $f_c$ strictly smaller than $0.1$ but still of the order of a few percent, see Fig. \[Z\_vs\_f\]. Although the value of $f_c$ is difficult to determine in general, we will assume in the following that such threshold probability exists and derive from it an expression for the memory lifetime.

![\[nonint\_T\] The values of $\tau$ extracted at the sharp transitions of the $\langle Z_{\rm ec} \rangle$ decay (circles). As in Fig. \[nonint\_fig\], we use $\gamma(0)=\gamma(2J)$. Comparison to 