26 FIG. 8: Scaling exponent for the variance of R$_{0}$ at various values of k . be restored once we take λ finite. It would require subtle cancellations to take place in order to do that. The upshot is that this is an interesting observation that the numerical exploration is giving us. It would be very hard to guess this behavior for observables with our current understanding of large N systems [45]. The one thing that is very special of this setup is that when we look at the eigenvalue density distribution in the saddle point limit, it is singular: it has a δ function behavior. Singular distributions get smeared by fluctuations and because they are singular, they might induce anomalous scalings of fluctuations, etc. Again, a thorough understanding of these issues requires further work. C. A scaling relation: size of the hole for varying q at fixed N The next item in the list is to check some of the geometric features of our ensemble and compare them to the expected solutions of supergravity and in particular with the free fermion picture [31, 40]. The idea is to fix N and vary Q , thereby exploring the parameter q = Q/N . We do this for our largest value of N = 10000 and for the largest k that we recorded k = 13. We plot R$_{0}$ ( q ) versus Q in the figure 9. A fit of the form R$_{0}$ ( q ) ∼ q α shows that the best fit is with α ≃ 0 . 47 ± 0 . 1, where the error is an estimate of the systematic error. We use the variance in the inner radius as a measurement of the systematic uncertainty. This is consistent with α = 0 . 5. Incidentally, α = 0 . 5 is the scaling value for the free fermion droplets. The reason for this is that the free fermion system (quantum hall system) is equivalent to a Coulomb gas in two dimensions. For that system both particles and holes form incompressible fluids, and the radius of the 0 2 4 6 8 10 12 0.2 0.4 0.6 0.8 1.0 1.2 1.4