$L \ge 16$. On moving to a finer $L=16$ lattice at $\beta=2.5$ the gulf was found to disappear and the origin of the long range physics was no longer so clear cut. This could be a mere finite volume effect, or, much more seriously, it might signal the breakdown of this monopole picture in the weak coupling, continuum limit. Clearly this needs to be resolved and the only unambiguous way to do so is by performing the calculations on large enough lattices.

This calculation {#ssec_thiscalc}
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The direct way to estimate the lattice size necessary at $\beta=2.5$ to restore (if that is possible) our picture is as follows. Suppose that the average size of the second largest cluster scales approximately as l\_ L\^()\^. We know that $l_{\max} \propto L^4 (\sqrt{K})^3$ to a good approximation for the largest cluster. So we will maintain the same ratio of lengths $l_{\second} / l_{\max}$, and a gulf between these, if = ( )\^[ -( )]{} \[eqn\_lK\_scale\] If we take our directly calculated values of $l_{\second}$, they seem to give roughly $\alpha \simeq 1$ and $\delta \simeq -2$. This suggests that we need to scale our lattice size with $\beta$ so as to keep $L(\sqrt{K})^{5/3}$ 