Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

By Francesco Guerra. 2002

629 citations recorded by the source corpus.

Abstract: By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick model, and the Derrida p-spin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from below by the value given in the Parisi Ansatz, uniformly in the size of the system. Moreover, the difference between the two expressions is given in the form of a sum rule, extending our previous work on the comparison between the true free energy and its replica symmetric Sherrington-Kirkpatrick approximation. We give also a variational bound for the infinite volume limit of the ground state energy per site.

Subfields: Condensed Matter Physics, Economics and Econometrics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e31b5b87e55f854afcb0fdd2c1e93c4a42328701

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Citations in the corpus

The parisi formula (2006, methodology): “We learned the present formulation in Guerra’s work [3], to which we refer for further discussion of its connections with Parisi’s original formulation.”
Hypothesis Testing in High-Dimensional Regression Under the Gaussian Random Design Model: Asymptotic Theory (2013, positive): “Attempts to make the replica method rigorous have been pursued over the last 30 years by some world-leading mathematicians [36, 37, 38, 39].”
The Parisi formula for mixed $p$-spin models (2011, background): “Guerra in [8] by inventing the replica symmetry breaking interpolation scheme.”
The Cavity Method at Zero Temperature (2002, methodology): “(2) The necessity of maximising over m is a well known feature of the replica method (in the limiting case k Q . it can be rigorously proven ( 32 ) that the value of F(m) is”
Fundamental limits of detection in the spiked Wigner model (2018, background): “One can show, following Guerra (2003), that there is a non-increasing sequence of thresholds ( λ k ) k ≥ 1 —each one corresponding to the point where the so-called “ k - RSB ” interpolation bound dips below zero—such that the above limit is strictly negative above λ ∞ = lim λ k .”
The mean field traveling salesman and related problems (2010, background): “Some important achievements in this direction are David Aldous’ proof of the ζ(2) limit in the assignment problem [4], Michel Talagrand’s proof of the correctness of the Parisi solution of the Sherrington–Kirkpatrick model [19], [34], [40], [44], and the algorithmic and theoretical results on phase ”
The Parisi ultrametricity conjecture (2011, background): “Guerra in [5], which gave a very strong indirect support to the entire Parisi ansatz including the ultrametricity conjecture.”