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

During the last few years, through the combined effort of the insight coming from physical intuition and computer simulation, and the exploitation of rigorous mathematical methods, the main features of the mean-field Sherrington–Kirkpatrick spin glass model have been firmly established. In particular, it has been possible to prove the existence and uniqueness of the infinite-volume limit for the free energy, and its Parisi expression, in terms of a variational principle involving a functional order parameter. Even the expected property of ultrametricity, 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 of spontaneous replica symmetry breaking (RSB), discovered by Parisi many years ago. By expanding on our previous work, the aim of this paper is to investigate a general framework, where replica symmetry breaking is embedded in a kind of mechanical scheme of the Hamilton–Jacobi type. Here, the analog of the 'time' variable is a parameter characterizing the strength of the interaction, while the 'space' variables rule out quantitatively the broken replica symmetry pattern. Starting from the simple cases, where annealing 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 the potential in the Hamilton–Jacobi equation as the level of symmetry breaking is increased. This new machinery allows us to work out mechanically the general K-step RSB solutions, in a different interpretation with respect to the replica trick, and easily reveals their properties such as existence or uniqueness.

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