The replica-symmetric free energy for Ising spin glasses with orthogonally invariant couplings
Zhou Fan, Yihong Wu
TL;DR
This work analyzes Ising spin glasses with orthogonally invariant couplings in a weak external field, establishing a replica-symmetric description of the first-order free energy at sufficiently high temperature. The authors adapt Bolthausen’s conditional second-moment method to a setting with orthogonal-invariance by employing a memory-free AMP algorithm to solve the TAP equations and by deriving sharp state-evolution and Haar-integration results for the resulting conditional moments. They construct explicit low-dimensional variational problems for both the conditioned first and second moments and prove that their optimizers converge to the replica-symmetric predictions, thereby validating the RS free energy formula in this broad class of models. The approach highlights universality phenomena for AMP dynamics under orthogonally invariant couplings and provides a rigorous spine for the TAP/RS correspondence beyond i.i.d. Gaussian couplings, with potential implications for statistical physics, information theory, and high-dimensional inference.
Abstract
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.