Replica symmetry breaking in mean-field spin glasses through the Hamilton–Jacobitechnique

Abstract

During the last few years, through the combined effort of the insight coming from physicalintuition and computer simulation, and the exploitation of rigorous mathematical methods,the main features of the mean-field Sherrington–Kirkpatrick spin glass model have beenfirmly established. In particular, it has been possible to prove the existence and uniquenessof the infinite-volume limit for the free energy, and its Parisi expression, in terms of avariational principle involving a functional order parameter. Even the expected property ofultrametricity, for the infinite-volume states, seems to be near to a complete proof.The main structural feature of this model, and related models, is the deep phenomenon ofspontaneous replica symmetry breaking (RSB), discovered by Parisi many years ago. Byexpanding on our previous work, the aim of this paper is to investigate a generalframework, where replica symmetry breaking is embedded in a kind of mechanical schemeof the Hamilton–Jacobi type. Here, the analog of the ‘time’ variable is a parametercharacterizing the strength of the interaction, while the ‘space’ variables rule outquantitatively the broken replica symmetry pattern. Starting from the simple cases, whereannealing is assumed, or replica symmetry, we build up a progression of dynamical systems,with an increasing number of space variables, which allow us to weaken the effect of thepotential in the Hamilton–Jacobi equation as the level of symmetry breaking is increased.This new machinery allows us to work out mechanically the generalK-step RSB solutions, in a different interpretation with respect to the replica trick, andeasily reveals their properties such as existence or uniqueness.