| Mean field theory is usually valid when the number of dimensions goes to infinity. I will show that the statistical mechanics of hard spheres (neglecting crystallization) can be exactly computed in this limit. The glass transition is present and the scenario introduced by Kirkpatrick, Thirumalai, and Wolynes is correct. We find that at high density in the glass phase the simplest solution becomes unstable and a full replica symmetry breaking phase is present. The fullRSB solution predicts correctly that jammed packings are isostatic, and allows one to compute analytically the critical exponents associated with the jamming transition, which are missed by the 1RSB solution. These predictions compare very well with numerical results, in a nearly dimension independent way. |