Thouless-Anderson-Palmer equations for the generic p-spin glass model

TL;DR

This paper proves that the TAP equations hold for the p-spin glass model on the hypercube under certain conditions, using a novel ultrametric decomposition of the Gibbs measure to analyze local spin behavior.

Contribution

It introduces a new ultrametric decomposition approach to establish the validity of TAP equations for generic p-spin glass models with a jump in the Parisi measure.

Findings

TAP equations hold for all sites with respect to conditional laws

Decomposition of Gibbs measure into a mixture of conditional laws

Applicable to models with a jump in the Parisi measure

Abstract

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.