for the one loop dilation operator (see Appendix A) is rather complicated. This is to be expected - it was obtained by summing a huge class of Feynman diagrams - much more than just the planar diagrams. On the other hand, the spectra of anomalous dimensions obtained are rather simple. To obtain a numerically tractable problem, we have been forced to keep the value of $a_{\rm max}$ finite (recall that this parameter measures the maximum value for the difference between the number of boxes in the long column and the number of boxes in the short column). In the large $N$ limit[^5] $a_{\rm max}=\zeta N$ goes to infinity. Our discussion assumes we are working in this $a_{\rm max}\to\infty$ limit.

We have restricted ourselves to a study of less than five impurities. This restriction is not imposed because our methods break down for five or more impurities, but only because the details of writing down the projection operators and evaluating the dilatation operator becomes increasingly complicated as the number of impurities is increased. In particular, there is no problem in principle with taking $O(N)$ impurities. The case of no impurities or one impurity are simple to handle analytically - all of 