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In the first part of this paper, we derive the general replica equations thatdescribe infinite-dimensional hard spheres at any level of replica symmetrybreaking (RSB) and in particular in the fullRSB scheme. We show that theseequations are formally very similar to the ones that have been derived for spinglass models, thus showing that the analogy between spin glasses and structuralglasses conjectured by Kirkpatrick, Thirumalai, and Wolynes is realized in astrong sense in the mean field limit. We also suggest how the computation couldbe generalized in an approximate way to finite dimensional hard spheres. In thesecond part of the paper, we discuss the solution of these equations and wederive from it a number of physical predictions. We show that, below theGardner transition where the 1RSB solution becomes unstable, a fullRSB phaseexists and we locate the boundary of the fullRSB phase. Most importantly, weshow that the fullRSB solution predicts correctly that jammed packings areisostatic, and allows one to compute analytically the critical exponentsassociated with the jamming transition, which are missed by the 1RSB solution.We show that these predictions compare very well with numerical results.

Exact theory of dense amorphous hard spheres in high dimension. III. The full RSB solution