Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

TL;DR

This paper proves that the Parisi Ansatz provides a lower bound for the free energy in mean field spin glass models, extending previous results to include full replica symmetry breaking and establishing a sum rule for the difference.

Contribution

It extends the interpolation method to compare the Parisi free energy with the true free energy, including full replica symmetry breaking, and derives a sum rule for their difference.

Findings

The quenched free energy is bounded from below by the Parisi expression.

The difference between the true free energy and the Parisi approximation is expressed as a sum rule.

A variational bound for the ground state energy per site is provided.

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.