Interpolating the Sherrington–Kirkpatrick replica trick

Abstract

Interpolation techniques have become, in the past decades, a powerful approach to describe several properties of spin glasses within a simple mathematical framework. Intrinsically, for their construction, these schemes were naturally implemented in the cavity field technique, or its variants such as stochastic stability and 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 apply the interpolation scheme to the original replica trick framework and test it directly on the cited paradigmatic model. Although the problem, at a mathematical level, has been deeply investigated by Talagrand, it is still rich in information from a theoretical physics perspective; in fact, by treating the number of replicas n ∈ (0, 1] as an interpolating parameter (far from its original interpretation) the proof of the attendant commutativity of the zero replica and the infinite volume limits can be easily obtained. Further, within this perspective, we can naturally think of n as a quenching temperature close to that introduced in off-equilibrium approaches to gain some new insight into our understanding of the off-equilibrium features encountered in equilibrium statistical mechanics of spin glasses.