An Autonomous Spectral Representation for Fully Asymmetric Spherical Sherrington-Kirkpatrick Dynamics

Source 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