20 FIG. 3: A log plot of the data in figure 2 A fit to log suggests that δf$_{k}$/f$_{k}$ ≃ Ak 0 . $^{35}$, but looking carefully at the data in figure 3 the power law behavior sets in after k = 5 and for low values of k the answer is different. Since we do not have a theory of these fluctuations, we can not do a fit that is sufficiently informed to know if this behavior is typical and represents a departure from the large N limit or not. Also, one show notice that since all of the variables we are measuring are strictly positive, the distribution of the variables is not gaussian. For δf$_{k}$/f$_{k}$ sufficiently small this should not matter too much. However for large k this is important and therefore the measurement of the fluctuations will have (large) non-gaussianities. Without a statistical model for how to deal with these issues, the numbers above just give order of magnitude estimates for deviations from typicality. Now, we can go to a more extreme case, where the 1 /N corrections are much larger. We do this in the figure 4, which is worked for the case N = 1000 , Q = 5. Notice that in this case the sizes of the different f$_{k}$ changes by various orders of magnitude at high k between different configurations. Hence the fluctuations are very large and dominate any calculation. This is a situation where we expect that the averages 〈 f$_{k}$ 〉 do not converge for k large, as described in the previous section. Hence the behavior of the graph is as would be expected: the large values of f$_{k}$ are dominated by large fluctuations rather than by the mean values. For the same extreme setup, we can also plot the radius as measured by the three different measurements that we suggested in the previous section. This appears in the figure 5. As can be read from the figure, we find that for various k , the definition of the radius R$_{0}$ converges faster than the definition of the radius based on the roots of f$_{k}$ . We can prove this in general (the argument is found below). Moreover, we see that the naive definition of the inner radius GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> GLYPH<SolidCircle> 2 4 6 8 10 12 14 0.001 0.002 0.005 0.010 0.020 0.050 0.100