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.