effects of the fluctuations of the number of anyons and their positions. These result in additional errors and correlated spin-flips across the memory, due to the long-range nature of the anyon interactions. Although we expect in general deviations from Eq. (\[T\_estimate\_int\]), the memory remains self-correcting both for an Ohmic and for a super-Ohmic bath.

Indeed, for an Ohmic bath, we can neglect the effect of the repulsive force if the change of energy $\omega$ in a diffusive step is smaller in magnitude than $T$ \[see Eq. (\[gamma\_bath\_text\])\], so that we can approximate $\gamma(\omega)\simeq \gamma(0)$. In particular, for a single pair of anyons at distance $r$, we have $|\omega| \lesssim \alpha A/r^{\alpha+1} $, which defines a critical radius $$r_c =( \alpha A\beta)^{\frac{1}{\alpha+1}},$$ beyond which the fluctuations become negligible. For $\alpha=0$ one has $r_c=0$. For $\alpha >0$, since the average distance $\sim 1/\sqrt{n_{\rm mf}}$ between anyons grows with $L$ while $r_c$ is independent of $L$, the fluctuations also become negligible. The validity of Eq. (\[T\_estimate\_int\]) for the Ohmic case is confirmed by numerical simulations both for $\alpha=0$ (see Fig. \[Zav\_90thresh\_int\]) and for $\alpha>0$ (see Fig. \[non\_constant\_interaction\]).

Concerning the super-Ohmic case, Eq. (\[T\_estimate\_int\]) could become inaccurate if the fluctuations of $\omega \simeq 0$ 