completely compelling of course. In the Gaussian approximation, for example, the $O(1/\beta)$ ultraviolet fluctuations would not generate any monopoles at all, and in that case there would be no reason to expect any breakdown of monopole dominance. Nonetheless the arguments do suggest that it would be surprising if the long distance physics were to be usefully and simply encoded in the monopole structure (as defined on the smallest ultraviolet scales) all the way to the continuum limit.

There are different ways in which monopole dominance could be lost. The most extreme possibility is that as $a \to 0$ the fields simply cease to contain monopole clusters that are large enough to disorder large Wilson loops. That this is indeed so has been argued in [@grady98] where it has been claimed that the exponent $\gamma$ in our eqn. \[eqn\_len\_spec\] (but defined for loops rather than for clusters) increases rapidly with decreasing $a$. Of course this would not in itself preclude the existence of a large percolating cluster, as long as this cluster could be decomposed into a large number of small and correlated intersecting loops. Irrespective of this, we also note that the volumes used in [@grady98] are very small by 