$$\label{T_estimate_int}
\tau \simeq \frac{2 f_{c} / n_{\rm mf}}{\max\{ \gamma(0),4\gamma(-2\epsilon_{\rm mf})\}}.$$ From this we obtain the lifetime of an interacting memory in case of an Ohmic ($n=1$) or super-Ohmic ($n > 1$) bath as $$\label{T_estimate_int1}
\tau \simeq
\begin{cases} 
\dfrac{f_c L_\alpha}{\kappa_1 T \ln L_\alpha} , & \text{Ohmic}
\vspace{0.3cm}
\\
\dfrac{2 f_{c}L_\alpha^3}{\kappa_n T^n (2\ln L_\alpha)^{n+3}} , &\text{super-Ohmic}
\end{cases}$$ in the limit of large grid size \[see after Eq. (\[leadingNmf\])\]. It is clear from these expressions that the memory lifetime is diverging with $L$, in strong contrast to the non-interacting case where it was bounded by a constant. In the Ohmic case, this divergence of $\tau$ is entirely due to the vanishing density, since $\gamma(0)= 2 \kappa_1 T$ is non-zero. In the super-Ohmic case, however, an additional divergence due to the vanishing of $\gamma(-2\epsilon_{\rm mf})$ is obtained, see Eq. (\[gamma\_2emf\]). Since the energy gap grows logarithmically with $L$, $\tau$ grows polynomially, but with a rather favorable power. For instance, constant interaction ($\alpha=0$, see also below) leads to $\tau \propto L^2/\ln L$ in the Ohmic case and to $\tau \propto L^6/\ln^5 L$ in the super-Ohmic ($n=2$) case.

Effects beyond the mean-field treatment
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Equation (\[T\_estimate\_int\]) is valid in the mean-field limit and does not include 