An Autonomous Spectral Representation for Fully Asymmetric Spherical Sherrington-Kirkpatrick Dynamics
Abstract
We study the large size dynamics of a fully asymmetric spherical Sherrington- Kirkpatrick model, allowing for a distinguished random signal that interacts with the evolving spin configuration. We first derive the large-N dynamical mean-field limit and characterize the law of a typical coordinate by an effective Gaussian process whose covariance must satisfy a self-consistency relation. The resulting formulation is intrinsically nonlocal in time and is not, in its usual form, a closed finite-dimensional dynamical system. Our main contribution is an autonomous spectral representation of this limiting dynamics. We introduce an auxiliary stationary chain of Gaussian replicas whose one-step marginal reproduces the effective field appearing in the dynamical mean- field equation. Applying a matrix-valued Herglotz representation, or equivalently Fourier analysis in the replica index, converts the covariance structure of the chain into a family of finite-dimensional matrix-valued evolution equations indexed by the unit circle. The macroscopic observables, including the signal overlap, are then recovered from explicit spectral integrals. This representation replaces the original two-time self-consistency problem by an autonomous spectral closure and provides a convenient framework for both analysis and numerical computation. We prove the validity of the representation and compare its predictions with direct simulations of the finite-dimensional system
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BibTeX
@misc{maclaurin2026autonomous,
title = {{An Autonomous Spectral Representation for Fully Asymmetric Spherical Sherrington-Kirkpatrick Dynamics}},
author = {James MacLaurin and Pedro Vilanova},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39115},
eprint = {2610.39115},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39115},
}
RIS
TY - UNPB
TI - An Autonomous Spectral Representation for Fully Asymmetric Spherical Sherrington-Kirkpatrick Dynamics
AU - James MacLaurin
AU - Pedro Vilanova
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39115
UR - https://arxiv.org/abs/2610.39115
AB - We study the large size dynamics of a fully asymmetric spherical Sherrington- Kirkpatrick model, allowing for a distinguished random signal that interacts with the evolving spin configuration. We first derive the large-N dynamical mean-field limit and characterize the law of a typical coordinate by an effective Gaussian process whose covariance must satisfy a self-consistency relation. The resulting formulation is intrinsically nonlocal in time and is not, in its usual form, a closed finite-dimensional dynamical system. Our main contribution is an autonomous spectral representation of this limiting dynamics. We introduce an auxiliary stationary chain of Gaussian replicas whose one-step marginal reproduces the effective field appearing in the dynamical mean- field equation. Applying a matrix-valued Herglotz representation, or equivalently Fourier analysis in the replica index, converts the covariance structure of the chain into a family of finite-dimensional matrix-valued evolution equations indexed by the unit circle. The macroscopic observables, including the signal overlap, are then recovered from explicit spectral integrals. This representation replaces the original two-time self-consistency problem by an autonomous spectral closure and provides a convenient framework for both analysis and numerical computation. We prove the validity of the representation and compare its predictions with direct simulations of the finite-dimensional system
ER -