Thouless et al. (Phys Mag 35(3):593–601, 1977), derived a representation for the free energy of the Sherrington–Kirkpatrick model, called the TAP free energy, written as the difference of the energy and entropy on the extended configuration space of local magnetizations with an Onsager correction term. In the setting of mixed p-spin models with Ising spins, we prove that the free energy can indeed be written as the supremum of the TAP free energy over the space of local magnetizations whose Edwards–Anderson order parameter (self-overlap) is to the right of the support of the Parisi measure. Furthermore, for generic mixed p-spin models, we prove that the free energy is equal to the TAP free energy evaluated on the local magnetization of any pure state.
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Acknowledgments. W.-K.C. thanks Antonio Auffinger, David Belius, and Nicola Kistler for fruitful discussions during the early stage of this work. Both authors thank the anonymous referees for the careful reading and providing several suggestions regarding the presentation of the paper. The research of W.-K.C. is partially supported by NSF DMS-1642207 and Hong Kong Research Grants Council GRF-14302515. D.P. is partially supported by NSERC.