High temperature TAP upper bound for the free energy of mean field spin glasses

Abstract: This work proves an upper bound for the free energy of the Sherrington-Kirkpatrick model and its generalizations in terms of the Thouless-Anderson-Palmer (TAP) energy. The result applies to models with spherical or Ising spins and any mixed $p$-spin Hamiltonian with external field or with a non-linear spike term. The bound is expected to be tight to leading order at high temperature, and is non-trivial in the presence of an external field. For the proof a geometric microcanonical method is employed, in which one covers the spin space with sets, each of which is centered at a magnetization vector $m$ and whose contribution to the partition function is bounded in terms of the TAP energy at $m$.