We derive the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics for spherical mixed p -spin disordered mean-field models, starting uniformly within one of the spherical bands on which the Gibbs measure concentrates at low temperature for the pure p -spin models and mixed perturbations of them. We further relate the large time asymptotics of the resulting coupled non-linear integro-differential equations, to the geometric structure of the Gibbs measures (at low temperature), and derive their FDT solution (at high temperature).

中文翻译：

---

球形自旋眼镜的动力学：无序依赖的初始条件

对于球形混合 p-spin 无序平均场模型，我们在 Langevin 动力学中推导出经验相关性和响应函数的热力学极限，均匀地开始于 Gibbs 测量在低温下集中于纯 p-的球带之一。自旋模型及其混合扰动。我们进一步将所得耦合非线性积分微分方程的大时间渐近性与 Gibbs 测度的几何结构（在低温下）联系起来，并推导出它们的 FDT 解（在高温下）。