Interpolating the Sherrington–Kirkpatrick replica trick

Abstract

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. Keywords: cavity methodspin glassesreplica trick Acknowledgments The strategy outlined in this paper is part of the study supported by the Italian Ministry for Education and Research, FIRB grant number RBFR08EKEV, and partially by Sapienza Università di Roma. FG is partially funded by INFN (Istituto Nazionale di Fisica Nucleare) which is also acknowledged. The authors are pleased to thank Dmitry Panchenko and an anonymous referee for their kind suggestions. We dedicate this paper to David Sherrington on the occasion of his seventieth birthday. Notes Notes 1. We learned this beautiful metaphor from Ton Coolen, whom we thank. 2. Here and in the following, we set the Boltzmann constant k B equal to one, so that β = 1/(k B T) = 1/T. 3. This procedure is deeply related to the mean field nature of the interactions, which ultimately allows one to consider even the low-temperature regimes as expressed in terms of high-temperature solutions 36 Talagrand, M. 2006. Ann. Math., 163: 221[Crossref], [Web of Science ®] , [Google Scholar]. 4. High temperature is the β-region where there is only one solution, i.e. q = 0, of the self-consistency relation. When this condition breaks down, a phase transition to a broken replica phase appears; we label β c that particular value of the temperature. 5. We allow ourselves a little abuse of notation in forgetting the β dependence for now. 6. Of course for simple systems, such as for instance the Curie–Weiss model where P(J) ∼ δ(J − 1), this term does not contribute to the thermodynamics and there is no n-dependence.

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