constant. This estimate is not entirely reliable because, on smaller lattices, the distributions of the ‘largest’ and ‘second largest’ clusters overlap so that they exchange [*rôles*]{}. An alternative estimate can be obtained from the tail of the distribution in eqn. \[eqn\_len\_spec\] that integrates to unity. Doing so [@hart98] one obtains $\alpha \simeq 2$ and $0 < \delta < 0.25 $. This suggests that we scale our lattice size so as to keep $L(\sqrt{K})^{\{1.4 \to 1.5\}}$ constant. This estimate is also not very reliable, since it assumes that the distribution of secondary cluster sizes on different field configurations fluctuates no more than mildly about the average distribution given in eqn. \[eqn\_len\_spec\]. In fact the fluctuations are very large. \[As we can see immediately when we try to calculate $\langle l^2 \rangle$ in order to obtain a standard fluctuation — it diverges for a length spectrum with $N(l)\propto dl/l^3$.\] Nonetheless, the two very different estimates we have given above produce a very similar final criterion: to maintain the same gap between the largest and second largest clusters as $\beta$ is varied, one should choose $L$ so as to keep $L(\sqrt{K})^{\sim 1.5}$ constant.

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