Dynamical mean-field limit and replica-symmetric free energy for the orthogonally-invariant SK model
Abstract
We study a class of diffusion processes on ℝn interacting through a symmetric matrix X∈ℝn× n. When eigenvectors of X are Haar-uniform on the orthogonal group, we derive a dynamical mean-field limit for the empirical law of sample paths, extending the classical Sompolinsky--Zippelius characterization for X∼GOE. The limit takes the form of a generalized Langevin equation with correlated Gaussian noise and memory, whose correlation and response kernels relate to those of the original dynamics through convolution equations involving the free cumulants of the eigenvalue distribution of X. For the overdamped Langevin diffusion associated with μ(θ)∝ exp (1/2θ⊤Xθ)∏i=1nν(dθi), we analyze the mean-field limit under a rapid-mixing assumption. The correlation and response kernels admit time-translation-invariant approximants satisfying a fluctuation-dissipation relation. The generalized Langevin equation admits a Markovian approximation coupled to an auxiliary multivariate OU process and converges to a replica-symmetric prediction for the empirical coordinate law under μ. This auxiliary correlation structure is characterized through the infinitesimal generator of a Markov semigroup for the lifted path-history process. Consequently, the free energy converges to a replica-symmetric limit under an explicit high-temperature condition, which for an Ising model is |X|op<1/2. By recent dynamical universality results, the same free-energy characterization holds for deterministic models without random disorder when X satisfies a set of deterministic delocalization conditions.
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BibTeX
@misc{fan2026dynamical,
title = {{Dynamical mean-field limit and replica-symmetric free energy for the orthogonally-invariant SK model}},
author = {Zhou Fan and Theodor Misiakiewicz and Leda Wang and Garrett G. Wen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.10102},
eprint = {2607.10102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.10102},
}
RIS
TY - UNPB
TI - Dynamical mean-field limit and replica-symmetric free energy for the orthogonally-invariant SK model
AU - Zhou Fan
AU - Theodor Misiakiewicz
AU - Leda Wang
AU - Garrett G. Wen
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2607.10102
UR - https://arxiv.org/abs/2607.10102
AB - We study a class of diffusion processes on ℝ^n interacting through a symmetric matrix X∈ℝ^n× n. When eigenvectors of X are Haar-uniform on the orthogonal group, we derive a dynamical mean-field limit for the empirical law of sample paths, extending the classical Sompolinsky--Zippelius characterization for X∼GOE. The limit takes the form of a generalized Langevin equation with correlated Gaussian noise and memory, whose correlation and response kernels relate to those of the original dynamics through convolution equations involving the free cumulants of the eigenvalue distribution of X. For the overdamped Langevin diffusion associated with μ(θ)∝ exp (1/2θ^⊤Xθ)∏_i=1^nν(dθ_i), we analyze the mean-field limit under a rapid-mixing assumption. The correlation and response kernels admit time-translation-invariant approximants satisfying a fluctuation-dissipation relation. The generalized Langevin equation admits a Markovian approximation coupled to an auxiliary multivariate OU process and converges to a replica-symmetric prediction for the empirical coordinate law under μ. This auxiliary correlation structure is characterized through the infinitesimal generator of a Markov semigroup for the lifted path-history process. Consequently, the free energy converges to a replica-symmetric limit under an explicit high-temperature condition, which for an Ising model is |X|_op<1/2. By recent dynamical universality results, the same free-energy characterization holds for deterministic models without random disorder when X satisfies a set of deterministic delocalization conditions.
ER -