The mean field theory of spin glasses: the heuristic replica approach and recent rigorous results

Abstract

The mathematically correct computation of the spin glasses free energy in the infinite range limit crowns 25 years of mathematic efforts in solving this model. The exact solution of the model was found many years ago by using a heuristic approach; the results coming from the heuristic approach were crucial in deriving the mathematical results. The mathematical tools used in the rigorous approach are quite different from those of the heuristic approach. In this note we will review the heuristic approach to spin glasses in the light of the rigorous results; we will also discuss some conjectures that may be useful to derive the solution of the model in an alternative way.

12 pages, 1 figure; lecture at the Flato Colloquia Day, Thursday 27 November, 2008

References in corpus (8)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• About the overlap distribution in mean field spin glass models
• Large Deviations in the Free-Energy of Mean-Field Spin-Glasses
• Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
• Replica symmetry breaking related to a general ultrametric space II: RSB solutions and the n\to0 limit
• Replica method and finite volume corrections

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• The free energy of a quantum Sherrington-Kirkpatrick spin-glass model for weak disorder