11 The simplest wave functions that change the topology are those that involve many (max- imal) giant gravitons on top of each other. The giant gravitons in the type IIB gravity theory in AdS$_{5}$ × S 5 are extended brane configuration with the same quantum numbers as gravitons [37] that wrap an S 3 inside S $^{5}$. If one places many of them on top of each other one expects a throat to form in the geometry near the location of the branes and to excise the branes from the geometry and replace them by fluxes [31, 38, 39]. The new topological feature is associated to the removal of the branes and the capping of the region of spacetime where they have been removed. In the dual field theory these are described at weak coupling in [32] and can be related to a system of free fermions on a two-dimensional phase space [33, 40]. This picture, in terms of incompressible quantum droplets is recovered geometrically in the half-BPS geometries [31], where the exact geometries of the capping are understood for half BPS D-branes with arbitrary shapes. These giant graviton states have been extensively studied at weak coupling. A recent review of many of these results can be found in [41]. Now, just having the map of droplets to states and to gravity does not mean that we understand the ten dimensional geometry from field theory. It just gives us a correspondence of particular solutions, and our task would be to explain the ten dimensional geometry from this data: how did the extra dimensions arise and why is there locality, time bending, etc., in the geometries. This is too hard with present techniques, which is why we will look at the field theory probability distributions for the quenched wave functions directly. The wave functions we consider are given by where Z 1 = X 1 + iX 2 in terms of the matrices and Q is an integer determining the number of giant gravitons. These are also given by z 1 i coordinates. Notice that Q is an integer in order for the wave function to be single valued. The normalization of ψ does not matter to determine the eigenvalue distribution, after all, wave functions are rays in a Hilbert space. The associated probability distribution of the eigenvalues (particles in six dimensions) is given by Notice that this probability distribution can be interpreted as a Boltzmann gas of particles with two body repulsive logarithmic interactions in 6 dimensions, subject to a background ψ$_{Q}$ = det( Z $^{1}$) $^{Q}$ψ$_{0}$ (6) p ∼ exp ( − ∑ i | ⃗ x$_{i}$ | 2 + Q ∑ i log( | z 1 i | $^{2}$) + ∑ i<j log | ⃗ x$_{i}$ − ⃗ x$_{j}$ | 2 ) (7)