The Annals of Probability

Thouless–Anderson–Palmer equations for generic $p$-spin glasses

Antonio Auffinger and Aukosh Jagannath

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We study the Thouless–Anderson–Palmer (TAP) equations for spin glasses on the hypercube. First, using a random, approximately ultrametric decomposition of the hypercube, we decompose the Gibbs measure, $\langle \cdot \rangle_{N}$, into a mixture of conditional laws, $\langle \cdot \rangle_{\alpha,N}$. We show that the TAP equations hold for the spin at any site with respect to $\langle \cdot \rangle_{\alpha,N}$ simultaneously for all $\alpha $. This result holds for generic models provided that the Parisi measure of the model has a jump at the top of its support.

Article information

Ann. Probab., Volume 47, Number 4 (2019), 2230-2256.

Received: May 2017
Revised: August 2018
First available in Project Euclid: 4 July 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses) 60G15: Gaussian processes

Spin glasses TAP random measures ultrametricity cluster decomposition


Auffinger, Antonio; Jagannath, Aukosh. Thouless–Anderson–Palmer equations for generic $p$-spin glasses. Ann. Probab. 47 (2019), no. 4, 2230--2256. doi:10.1214/18-AOP1307.

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