16 R$_{0}$ = lim k →∞ 〈 ( r 2 k +2 k +1 r 2 k k ) 1 / 2 〉 = lim k →∞ 〈 ( f$_{k}$ f$_{k}$$_{+1}$ ) 1 / 2 〉 (19) R 2 0 | N →∞ = lim k →∞ 〈 f$_{k}$ f$_{k}$$_{+1}$ 〉 = lim k →∞ 〈 f$_{k}$ 〉 〈 f$_{k}$$_{+1}$ 〉 (20) study the result for various k and collect them together. We should see convergence to a limit in the numerical exploration of these configurations. For our setups the k will vary between one and fourteen. If the inner radius is at 1 / 2 of the full size of the geometry, the suppression of the outer edge contribution relative to the inner edge in a given sum can be as large as (0 . 5) 28 ≃ 10 − 9 for the case of maximum k , but for particles sitting at r ≃ 1 . 1 r$_{in}$ the suppression is only a factor of 14 smaller, so they should still be contributing. Of course, if the inner radius is very small, the near inner radius region is also very small and the suppression of the other regions will be more pronounced. There is a third definition that is useful, and that is the following This is another way of removing the problem with the poles, and instead of a root of the single trace expressions, we are taking a ratio of these expressions for different k . In the ratio, if the numerator gets big when one particle goes towards the z 1 locus, the denominator also gets large at the same time, and in the observables the average is not unduly dominated by large fluctuations. It turns out that this last definition, from the ones we have given, is optimal. We will explain how this works in the next section, where we can see directly in the data why this is so. Notice moreover, that if we take N → ∞ first, one can use large N factorization (one expects small corrections in inverse powers of N ), so in the limit we also have that so it is clear that the different definitions are not too different from each other at large N . However, they behave very differently at finite N . All of these functions are chosen because when we take N → ∞ at q fixed, they have (at least in principle) a theoretically controlled approach to large N that can be evaluated around the saddle point configuration. There are other reasons to choose inverse powers of the radius: in the paper of Chen et al. [43] it was shown that inverse powers of these matrices also appear in the description of closed string states in the presence of backgrounds with topology changes. Also, the moments of the eigenvalue density distribution serve as a characteristic description of shape that determines the half BPS states, and these can also be measured gravitationally in the AdS dual geometry [44] ( these particular calculations