by an asterisk), we obtain $\langle  Z \rangle= (1-
1/L)^{N^\ast/2} \simeq e^{-N^\ast/2L}$. By using $N^\ast \simeq 4
L^2 \gamma(-2J) t$, it follows that $\langle  Z \rangle$ decays exponentially with $L$.

For the interacting case, we find good agreement of a modified version of Eq. (\[T\_estimate\]) with the simulations \[see Eq. (\[T\_estimate\_int\]) and Fig. \[Zav\_90thresh\_int\]\]. Fitting the data always yields values of $f_c$ smaller than $f_c = 0.11$, but still of the order of a few percent. These values are thus consistent with the original meaning of $f_c$. For a more extended discussion, we refer to Secs. \[sec:lifetime\_mf\] and \[sec:lifetime\_interacting\].

Mean-field analysis of the interacting model {#sec:mf}
============================================

We now turn to the interacting case $A > 0$ and perform a mean-field analysis, which becomes accurate in the relevant limit of large $L$.

Mean-field anyon density
------------------------

We first consider the equilibrium number of anyons $N$ within a mean-field treatment (mean-field values will be indexed with a subscript ‘mf’). We obtain the single-particle energy at plaquette $p$ as $\epsilon_p=\delta H_0/\delta n_p= J+ \sum_{p'\neq p} U_{p p'} n_{p'}$. Replacing $n_{p'}$ by the average value $n_{\rm mf}=N_{\rm mf}/L^2$ and taking the continuum limit, we find the mean-field value for $\epsilon_p$ to be $$\label{Nmf}
