Approximating the Gibbs Measure of Spherical Spin Glasses to Within $o(N^{-1/2})$ in the Replica Symmetric Phase

Abstract

We study the magnetization vector of mixed spherical spin glasses in a weak external field, in the replica symmetric phase. We derive a coordinatewise expansion of the magnetization which includes the leading cubic correction to the quadratic approximation, with an error that is $o(N^{-1/2})$ per coordinate. We also prove a corresponding high-moment bound for the full vector. Finally, our companion paper applies these estimates to the effective Hamiltonian obtained by restricting a tilted spherical spin glass to a narrow band around a TAP critical point. The resulting sharp approximation is designed for applications in which normalized Wasserstein accuracy is insufficient, including total-variation sampling by stochastic localization.