Exact theory of dense amorphous hard spheres in high dimension. III: The full replica symmetry breaking solution. (English) Zbl 1456.82236
Summary: In the first part of this paper, we derive the general replica equations that describe infinite-dimensional hard spheres at any level of replica symmetry breaking (RSB) and in particular in the fullRSB scheme. We show that these equations are formally very similar to the ones that have been derived for spin glass models, thus showing that the analogy between spin glasses and structural glasses conjectured by Kirkpatrick, Thirumalai and Wolynes is realized in a strong sense in the mean-field limit. We also suggest how the computation could be generalized in an approximate way to finite-dimensional hard spheres. In the second part of the paper, we discuss the solution of these equations and we derive from it a number of physical predictions. We show that, below the Gardner transition where the 1RSB solution becomes unstable, a fullRSB phase exists and we locate the boundary of the fullRSB phase. Most importantly, we show that 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. We show that these predictions compare very well with numerical results.
For part I, see [J. Kurchan et al., J. Stat. Mech. Theory Exp. 2012, No. 10, Article ID P10012, 20 p. (2012; Zbl 1456.82284)].
For part I, see [J. Kurchan et al., J. Stat. Mech. Theory Exp. 2012, No. 10, Article ID P10012, 20 p. (2012; Zbl 1456.82284)].
MSC:
| 82B23 | Exactly solvable models; Bethe ansatz |
| 82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |
Citations:
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