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

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.