Remarks on the Interpolation Method
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Remarks on the Interpolation Method Roberto Boccagna1 · Davide Gabrielli1 Received: 1 April 2020 / Accepted: 8 August 2020 © The Author(s) 2020
Abstract We discuss a generalization of the classic condition of validity of the interpolation method for the density of quenched free energy of mean field spin glasses. The condition is written just in terms of the L 2 metric structure of the Gaussian random variables. As an example of application we deduce the existence of the thermodynamic limit for a GREM model with infinite branches for which the classic conditions of validity fail. We underline the dependence of the density of quenched free energy just on the metric structure and discuss the models from a metric viewpoint. Keywords Spin glasses · Interpolation method · Thermodynamic limit Mathematics Subject Classification 60G15 · 82D30
1 Introduction The interpolation method is a simple but powerful technique used to prove inequalities for Gaussian random vectors (see for example [20] and [21]). This method has great relevance in the field of Mathematical and Theoretical Physics since it represents an essential ingredient in the study of mean field spin glasses. In the breakthrough paper [19] it has been used to prove the existence of the thermodynamic limit for the quenched density of free energy for the Sherrington–Kirkpatrick model. This was a longstanding problem and its solution was the turning point towards the proof of the Parisi Formula [29]. Spin glasses are simple mathematical models for disordered systems whose rigorous analysis is indeed a challenge for mathematicians. We refer to [24,28] the mathematically interested reader and to [23] the physically interested one. Among plenty of models, one of the most studied is that introduced by Sherrington and Kirkpatrick in [26] as a solvable elementary model. Indeed the structure of the solution turned out to be much more rich and complex than expected and was build up in a series of papers by Parisi (see [23] for a detailed
Communicated by Federico Ricci-Tersenghi.
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Davide Gabrielli [email protected] Università dell’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila, Italy
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R. Boccagna, D. Gabrielli
discussion). A rigorous proof of the Parisi conjectured solution was missing for a long time and the interpolation method played a key role in its proof. See [18] for a review on this. Using the same idea of [19], the authors of [12] proposed a general setting for the interpolation method in the framework of mean field spin glasses. Furthermore, they successfully applied this technique to prove the existence of the thermodynamic limit for the Generalized Random Energy Model (GREM, a family of models introduced in [15]) with a finite number of levels. The interpolation method is now a powerful technique that has many different applications in different contexts, see for example [1–6,22], a list that is by far not exaustive. The “classical” hypothesis under which the interpolation method can be applied to the quenched free energy of mean field sp
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