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Dynamics for spherical spin glasses: Gibbs distributed initial conditions

Amir Dembo, Eliran Subag

TL;DR

This work derives the macroscopic Langevin-dynamics description for spherical mixed $p$-spin glasses initialized from Gibbs measures, providing coupled integro-differential equations for the empirical covariance $C$ and integrated response $oldsymbol{ ho}$ in both RS and $1$-RSB phases. By conditioning on the Gibbs initialization and employing a finite-mixture reduction, the authors obtain a rigorous framework in which stationary FDT behavior emerges above the dynamical transition, while 1RSB Gibbs-band localization governs low-temperature dynamics, including potential aging and no-aging regimes. They construct and analyze limiting dynamics ${ m U}^{ m sp}_{q_ullet}(oldsymbol{V})$ and ${ m U}^{f}_{q_ullet}(oldsymbol{V})$, prove existence, uniqueness, and continuity, and establish that infinite-mixture limits are approachable via finite-mixture approximations. The results provide a rigorous link between dynamical mean-field theory predictions (CKCHS equations) and Gibbs-initialized spin-glass dynamics, clarifying band-relaxation mechanisms, aging phenomena, and the role of $1$-RSB structure in non-equilibrium evolution, with implications for understanding slow relaxation in high-dimensional disordered systems.

Abstract

We derive the coupled non-linear integro-differential equations for the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics at temperature $T$, for spherical mixed $p$-spin disordered mean-field models, initialized according to a Gibbs measure for temperature $T_0$, in the replica-symmetric (RS) or $1$-replica-symmetry-breaking (RSB) phase. For any $T_0=T$ above the dynamical phase transition point $T_c^{\rm dyn}$ the resulting stationary relaxation dynamics coincide with the FDT solution for these equations, while for lower $T_0=T$ in the $1$-RSB phase, the relaxation dynamics coincides with the FDT solution, now concentrated on the single spherical band within the Gibbs measure's support on which the initial point lies.

Dynamics for spherical spin glasses: Gibbs distributed initial conditions

TL;DR

This work derives the macroscopic Langevin-dynamics description for spherical mixed -spin glasses initialized from Gibbs measures, providing coupled integro-differential equations for the empirical covariance and integrated response in both RS and -RSB phases. By conditioning on the Gibbs initialization and employing a finite-mixture reduction, the authors obtain a rigorous framework in which stationary FDT behavior emerges above the dynamical transition, while 1RSB Gibbs-band localization governs low-temperature dynamics, including potential aging and no-aging regimes. They construct and analyze limiting dynamics and , prove existence, uniqueness, and continuity, and establish that infinite-mixture limits are approachable via finite-mixture approximations. The results provide a rigorous link between dynamical mean-field theory predictions (CKCHS equations) and Gibbs-initialized spin-glass dynamics, clarifying band-relaxation mechanisms, aging phenomena, and the role of -RSB structure in non-equilibrium evolution, with implications for understanding slow relaxation in high-dimensional disordered systems.

Abstract

We derive the coupled non-linear integro-differential equations for the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics at temperature , for spherical mixed -spin disordered mean-field models, initialized according to a Gibbs measure for temperature , in the replica-symmetric (RS) or -replica-symmetry-breaking (RSB) phase. For any above the dynamical phase transition point the resulting stationary relaxation dynamics coincide with the FDT solution for these equations, while for lower in the -RSB phase, the relaxation dynamics coincides with the FDT solution, now concentrated on the single spherical band within the Gibbs measure's support on which the initial point lies.

Paper Structure

This paper contains 10 sections, 20 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Gibbs distributed $\mathbf{x}_0$ and conditional disorder
  3. Stationarity, challenges and organization
  4. Stationary dynamics, band relaxation, fdt regime and localized states with no-aging
  5. The fdt regime
  6. Localized states with no-aging
  7. Existence, confinement and continuity
  8. Proof of Proposition \ref{['prop-macro-easy']}
  9. Limiting dynamics: uniqueness and continuity
  10. Conditional mean and Hessian norm: Proof of Lemma \ref{['lem:G-conc']}

Key Result

Theorem 1.6

Fix finite $\beta,T>0$. Choose $\mathbf{x}_0 \in {\mathbb S}_N$ according to $\widetilde{\mu}^N_{2\beta_0,{\bf J}}$ and consider the dynamics diffusion, where the derivative of $f(\cdot)$ of the form eq:fdef is locally Lipschitz on $[0,r_\star^2)$ and the mixture $\nu(\cdot)$ satisfies eq:r-star. (a at an exponential in $N$ rate. (b). Suppose generic $\nu(\cdot)$ is strictly 1-rsb at $\beta_0 > \b

Theorems & Definitions (59)

  • Definition 1.1: $1$-rsb dynamics
  • Remark 1.2: singularity
  • Definition 1.3: rs dynamics
  • Remark 1.4: non-negative definiteness
  • Definition 1.5: Gibbs measure initialization
  • Theorem 1.6
  • Proposition 1.7
  • Remark 1.8: non-generic and $k$-rsb, $k \ge 2$
  • Definition 1.9
  • Theorem 1.10
  • ...and 49 more