define the bonds required for periodicity by tiling the plane with this cluster, as shown in Fig. \[11site\].

We study clusters of 8 to 12 sites on the square lattice. For $N=12$, there are about 1.7 $\times 10^7$ configurations and about 3.9 $\times 10^8$ transitions; restrictions of computer time and storage prevent us from going beyond this size. The crossings of $m$ between successive sizes do not yield useful predictions for $\lambda_c$ in this case. Evidently, the linear extent of the clusters is too small to probe the scaling regime. We instead derive estimates for the critical point by locating the maximum of $d\rho/d\lambda$, since in the infinite-size limit, this derivative (taken from the left) diverges at the critical point. The resulting predictions, for clusters of 11 and 12 sites, are compared with simulation in Fig. \[lm2d\], showing that the QS analysis provides a semiquantitative prediction for $\lambda_c$, and captures the shape of the phase boundary. (The simulation data in Fig. 5 are obtained by extrapolating moment ratio crossings [@moments] for system with linear sizes $L=40$, 80, 160 and 320). This analysis suggests that the phase transition is continuous (as found in simulation) since the QS probability distribution is unimodal 