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Central limit theorems for cavity and local fields of the Sherrington-Kirkpatrick model. (English) Zbl 1288.60126
The purpose of this paper is to derive central limit theorems (CLT) for cavity and local fields of the Sherrington-Kirkpatrick (SK) model in the high temperature regime. The cavity field plays an important role in the cavity method, which was used by Talagrand to prove the validity of the Thouless-Anderson-Palmer (TAP) equations. Talagrand proved that the limit law of the centered (with respect to Gibbs average) cavity field is that of a centered Gaussian distribution (note that the correct reference for Theorem 1.1 is Theorem 1.7.11 in [M. Talagrand, Mean field models for spin glasses. Volume I: Basic examples. Berlin: Springer (2011; Zbl 1214.82002)]) and raised the question of getting quantitative results for the limit law of the (non-centered) cavity field. Chatterjee obtained a first quantitative result using Stein’s method. Using Gaussian interpolation techniques, the author proves the CLT for the cavity field (Theorem 1.2) and the local field (Theorem 1.3) by means of quantitative moment estimates, and provides an application to the TAP equations (Corollary 1.4). Section 1 contains an introduction to the SK model and the TAP equations, and presents the main results, while Section 2 contains the proofs of Theorems 1.2 and 1.3.
MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60F05 Central limit and other weak theorems
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