Approximating the Gibbs Measure of Spherical Spin Glasses to Within $o(N^{-1/2})$ in the Replica Symmetric Phase
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Institutions: Stanford University, University of Chicago
Language: English
Subfield: Condensed Matter Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
Awards: CAREER: Phase Transitions in Randomized Combinatorial Search and Optimization Problems, MFAI: Computational and Statistical Limits in Generative Sampling, Collaborative Research: MFAI:Random geometry of the empirical risk landscape in deep learning: Optimization and Generalization, CAREER: Fundamentals of Random Discrete Systems
OpenAlex ID: W7223687911