3 quantum mechanics [15, 16], but they suffer from similar limits on taking N large. The main problem at hand is that in order to compare gravity and the field theory, in the regime where gravity is described by a semiclassical expansion, we are required to solve a field theory at strong coupling. This is extremely hard in general and this is why numerical methods might offer a good insight into the strong coupling behavior of these field theories. However, as is well known, the gravitational limit also requires solving the field theory at large N . Current lattice methods can in principle solve four dimensional theories for moderate values oF N ∼ 10, but taking N large is too difficult. Remember that in the AdS/CFT correspondence the radius of the five sphere and also of the AdS sphere in Planck units scales like N 1 / $^{4}$. Thus, in order to have a resolution of the sphere which is many times the Planck length, let us say by an order of magnitude, we require four orders of magnitude on N . Such numbers are beyond what can be done with a full simulation. Clearly, some approximation to the full field theory problem is required that reduces effectively the number of degrees of freedom to something manageable. Only at this stage simulations become a viable option to analyze the problem. Progress in this direction was made in [5]. It was argued there that at strong coupling, when the N = 4 SYM field theory is compactified on a three-sphere, the configurations of commuting (constant) scalar matrices dominate the infrared dynamics. Toy models that exhibit this property were exhibited in [17], giving additional indirect evidence for this proposal in the case of N = 4 SYM. The nature of the argument that leads to this conclusion requires various steps. First one argues that such classes of configurations are required for describing some supersymmetric states, so the truncation to such variables might be better behaved than naively expected. Afterwards one argues that the quartic scalar potential of the theory should be minimized. This second step is accomplished by systems of commuting matrices. The dominance of this configurations dynamically quenches the other off-diagonal degrees of freedom, meaning that they become heavy and their contributions can be ignored to some degree. The end product is a model describing only a handful of the degrees of freedom of the full N = 4 SYM theory, and since the other degrees of freedom are essentially ignored, we can call this setup a quenched minisuperspace approximation. We will use the name quenched in the rest of the paper. What has been found is that the distribution of eigenvalues of the commuting matrices (in this approximation) has a geometry of its own [5, 18]. The