Abstract
We introduce some applications of Stein’s method in the high temperature analysis of spin glasses. Stein’s method allows the direct analysis of the Gibbs measure without having to eate a cavity. Another advantage is that it gives limit theorems with total variation error bounds, although the bounds can be suboptimal. A surprising byproduct of our analysis is a relatively transparent explanation of the Thouless–Anderson–Palmer system of equations. Along the way, we develop Stein’s method for mixtures of two Gaussian densities.
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Acknowledgments
The author thanks Michel Talagrand, Persi Diaconis and the associate editor for various helpful suggestions. The author is also grateful to the referee for a very careful reading of the proofs and a large number of useful comments.
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The author’s research was partially supported by NSF grant DMS-0707054 and a Sloan Research Fellowship.
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Chatterjee, S. Spin glasses and Stein’s method. Probab. Theory Relat. Fields 148, 567–600 (2010). https://doi.org/10.1007/s00440-009-0240-8
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DOI: https://doi.org/10.1007/s00440-009-0240-8