The Replica-Symmetric Free Energy for Ising Spin Glasses with Orthogonally Invariant Couplings

We study a variant of the Sherrington-Kirkpatrick (S-K) spin glass model withexternal field, where the random symmetric couplings matrix does not consist ofi.i.d. entries but is instead orthogonally invariant in law. For sufficientlyhigh temperature, we prove a replica-symmetric formula for the first-orderlimit of the model free energy. Our analysis is an adaptation of a conditionalsecond-moment-method argument previously introduced by Bolthausen for studyingthe high-temperature regime of the S-K model, where one conditions on theiterates of an Approximate Message Passing (AMP) algorithm for solving the TAPequations for the model magnetization. We apply this method using a memory-freeversion of AMP that is tailored to the orthogonally invariant structure of themodel couplings.