We explore Langevin dynamics in the spherical Sherrington-Kirkpatrick model, delving into the asymptotic energy limit. Our approach involves integro-differential equations, incorporating the Crisanti-Horner-Sommers-Cugliandolo-Kurchan equation from spin glass literature, to analyze the system's size and its temperature-dependent phase transition. Additionally, we conduct an average case complexity analysis, establishing hitting time bounds for the bottom eigenvector of a Wigner matrix. Our investigation also includes the power iteration algorithm, examining its average case complexity in identifying the top eigenvector overlap, with comprehensive complexity bounds.

翻译：我们探索了球面Sherrington-Kirkpatrick模型中的朗之万动力学，深入研究了渐近能量极限。我们的方法涉及积分微分方程，融入了自旋玻璃文献中的Crisanti-Horner-Sommers-Cugliandolo-Kurchan方程，以分析系统尺度及其温度依赖的相变。此外，我们进行了平均情况复杂度分析，建立了Wigner矩阵底部特征向量的命中时间界。我们的研究还包括幂迭代算法，考察其在识别顶部特征向量重叠时的平均情况复杂度，并给出了全面的复杂度界限。