Interpolating the Sherrington-Kirkpatrick replica trick

TL;DR

This paper introduces an interpolation scheme applied to the replica trick for the Sherrington-Kirkpatrick spin glass model, enabling easier derivation of replica-symmetric and full RSB solutions, and establishing limit commutativity.

Contribution

It develops a novel interpolation approach within the replica trick framework, providing new insights into the SK model's solution structure and limit behaviors.

Findings

Derivation of replica-symmetric control using interpolation.

Description of the full RSB scenario with the new method.

Proof of the commutativity of zero replica and infinite volume limits.

Abstract

The interpolation techniques have become, in the past decades, a powerful approach to lighten several properties of spin glasses within a simple mathematical framework. Intrinsically, for their construction, these schemes were naturally implemented into the cavity field technique, or its variants as the stochastic stability or the random overlap structures. However the first and most famous approach to mean field statistical mechanics with quenched disorder is the replica trick. Among the models where these methods have been used (namely, dealing with frustration and complexity), probably the best known is the Sherrington-Kirkpatrick spin glass: In this paper we are pleased to apply the interpolation scheme to the replica trick framework and test it directly to the cited paradigmatic model: interestingly this allows to obtain easily the replica-symmetric control and, synergically with the broken replica bounds, a description of the full RSB scenario, both coupled with several minor theorems. Furthermore, by treating the amount of replicas $n\in(0,1]$ as an interpolating parameter (far from its original interpretation) this can be though of as a quenching temperature close to the one introduce in off-equilibrium approaches and, within this viewpoint, the proof of the attended commutativity of the zero replica and the infinite volume limits can be obtained.