  We study a variant of the Sherrington-Kirkpatrick (S-K) spin glass model with
external field, where the random symmetric couplings matrix does not consist of
i.i.d. entries but is instead orthogonally invariant in law. For sufficiently
high temperature, we prove a replica-symmetric formula for the first-order
limit of the model free energy. Our analysis is an adaptation of a conditional
second-moment-method argument previously introduced by Bolthausen for studying
the high-temperature regime of the S-K model, where one conditions on the
iterates of an Approximate Message Passing (AMP) algorithm for solving the TAP
equations for the model magnetization. We apply this method using a memory-free
version of AMP that is tailored to the orthogonally invariant structure of the
model couplings.
