Emergence of near-TAP free energy functional in the SK model at high temperature

Abstract

We study the SK model at inverse temperature $\beta >0$ and strictly positive field $h>0$ in the region of $(\beta ,h)$ where the replica-symmetric formula is valid. An integral representation of the partition function derived from the Hubbard-Stratonovitch transformation combined with a duality formula is used to prove that the infinite volume free energy of the SK model can be expressed as a variational formula on the space of magnetisations, $m$. The resulting free energy functional differs from that of Thouless, Anderson and Palmer (TAP) by the term $-\frac{{\beta }^2}{4}{\left(q-q_{\text{EA}}(m)\right)}^2$ where $q_{\text{EA}}(m)$ is the Edwards-Anderson parameter and $q$ is the minimiser of the replica-symmetric formula. Thus, both functionals have the same critical points and take the same value on the subspace of magnetisations satisfying $q_{\text{EA}}(m)=q$. This result is based on an in-depth study of the global maximum of this near-TAP free energy functional using Bolthausen's solutions of the TAP equations, Bandeira & van Handel's bounds on the spectral norm of non-homogeneous Wigner-type random matrices, and Gaussian comparison techniques. It holds for $(\beta ,h)$ in a large subregion of the de Almeida and Thouless high-temperature stability region.

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