We study the Langevin dynamics for spherical ‐spin models, focusing on the short time regime described by the Cugliandolo–Kurchan equations. Confirming a prediction of Cugliandolo and Kurchan, we show the asymptotic energy achieved is exactly in the low temperature limit. The upper bound uses hardness results for Lipschitz optimization algorithms and applies for all temperatures. For the lower bound, we prove the dynamics reaches and stays above the lowest energy of any approximate local maximum. In fact the latter behavior holds for any Hamiltonian obeying natural smoothness estimates, even with disorder‐dependent initialization and on exponential time‐scales.

中文翻译：

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纯球形自旋玻璃的低温朗之万动力学阈值能量

我们研究球自旋模型的朗之万动力学，重点关注 Cugliandolo-Kurchan 方程描述的短时状态。证实了 Cugliandolo 和 Kurchan 的预测，我们表明所达到的渐近能量恰好在低温极限内。上限使用 Lipschitz 优化算法的硬度结果并适用于所有温度。对于下界，我们证明动力学达到并保持在任何能量的最低能量之上近似局部最大值。事实上，后一种行为适用于任何服从自然平滑估计的哈密顿量，即使具有无序相关的初始化和指数时间尺度。