are more effective for the anyon motion than the indirect diffusion mechanism which is proportional to the rate in Eq. (\[gamma\_2emf\]). However, due to the decreasing interaction strength, such fluctuations in $\omega$ become small at large $L$ and still result in a vanishing diffusion coefficient. Therefore, Eq. (\[T\_estimate\_int\]) might overestimate the lifetime in this case, but the asymptotic dependence on $L$ would still be better than in the Ohmic case. Furthermore, at $\alpha=0$ direct hopping is impossible and Eq. (\[T\_estimate\_int\]) is valid (see Fig. \[Zav\_90thresh\_int\]).

Dynamics of the interacting model {#sec:lifetime_interacting}
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We turn now to the numerical simulations of our model, Eq. (\[H0\]), and focus first on constant long-range interactions ($\alpha=0$). In this case, the total energy $E_N= N J +\frac{A}{2}N(N-1)$ depends only on the number of anyons $N$, but not on their position. This simplifies the numerical treatment considerably. Our results are displayed in Fig. \[Zav\_90thresh\_int\]. The numerical data show a clear increase of the memory lifetime $\tau$ with $L$. Note that this holds already for the *bare* logical $Z$ operator. Like in the non-interacting case (see Fig. \[nonint\_fig\]), the beneficial effect of the error correction at read-out is to prolong the lifetime by maintaining $\langle Z_{\rm ec} 