used to study the accuracy of the mean-field value $N_{\rm mf}=n_{\rm mf}L^2$ \[see Eq. (\[meanfield\])\], in particular for values $\alpha \neq 0$. Due to the long-range nature of the interaction, $N_{\rm mf}$ compares very well to the equilibrium value of $N$ obtained from these simulations at generic values of the temperature and interaction exponent $\alpha$. This is illustrated in Fig. \[N\_metropolis\], which further shows a satisfactory agreement already at moderate values of $L$.

![\[N\_metropolis\] (Color online.) Comparison of the equilibrium value of $N$ obtained numerically (crosses) with $N_{\rm mf}$ (curves) for different grid sizes. We have used the interaction exponents $\alpha = 0$ (solid line), $\alpha = 0.5$ (dashed line), and $\alpha = 1.0$ (dotted line), and the temperature $T/J = 0.5$.](N_int_equilibrium.pdf){width="0.95\columnwidth"}

We also note that, for the case of constant interaction ($\alpha=0$), the average number can be calculated directly from the grand-canonical partition function $$\label{part_funct}
{\sum_{2k\leq L^2}}\left (
\begin{array}{c}
L^2 \\
2k
\end{array}
\right) e^{-\beta E_{2k}},$$ since the energy of a given anyon configuration does not depend on the positions of the anyons, but only on their total number $N=\sum_p n_p$. In the presence of a sufficiently strong anyon interaction or at low temperature, the number of excited plaquettes 