Exact theory of dense amorphous hard spheres in high dimension. III. The full replica symmetry breaking solution

By Patrick Charbonneau, Jorge Kurchan, Giorgio Parisi, Pierfrancesco Urbani, Francesco Zamponi. 2013

128 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Materials Chemistry, Probabilistic & Bayesian Methods

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Citations in the corpus

The jamming transition as a paradigm to understand the loss landscape of deep neural networks (2018, background): “Calculations in infinite dimensions [45, 60] showed that marginal stability is associated with a hierarchical organization of minima of the energy (a phenomenon referred to as replica symmetry breaking [61]), and exponents were found to follow γ = 0.”
Universality of the SAT-UNSAT (jamming) threshold in non-convex continuous constraint satisfaction problems (2017, methodology): “The scaling analysis is carried out along the lines of [ 13,14,38 ] .”
Edwards Statistical Mechanics for Jammed Granular Matter (2016, methodology): “We have reviewed the recent encouraging results of (Charbonneau et al., 2014b; Martiniani et al., 2017; Newman and Stein, 1999; Sharma et al., 2016) and have proposed a calculation for the 1-SF states in the SK model.”
Universal spectrum of normal modes in low-temperature glasses (2015, background): “, independent of the spatial dimension d for all d≥ 2 (26, 27), (ii) they can be analytically predicted through the exact solution in d→∞ (17, 28), and (iii) the perceptron model of neural networks, a prototypical CCSP, displays a jamming transition with the same exponents (25).”
Mean-field avalanches in jammed spheres. (2016, background): “tions in three dimensions. As stated in the introduction, at zero temperature one nds dierent solutions and scalings exactly at jamming ˚= ˚ J and in the soft sphere region ˚>˚ J. As described in [25] the jamming solution is characterized by a singular y(q) ˘(1 q) for q!1, where is related to th”