Majumdar Satya N, Vergassola Massimo
Laboratoire de Physique Théorique et Modèles Statistiques (UMR 8626 du CNRS), Université Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France.
Phys Rev Lett. 2009 Feb 13;102(6):060601. doi: 10.1103/PhysRevLett.102.060601. Epub 2009 Feb 12.
We present a Coulomb gas method to calculate analytically the probability of rare events where the maximum eigenvalue of a random matrix is much larger than its typical value. The large deviation function that characterizes this probability is computed explicitly for Wishart and Gaussian ensembles. The method is general and applies to other related problems, e.g., the joint large deviation function for large fluctuations of top eigenvalues. Our results are relevant to widely employed data compression techniques, namely, the principal components analysis. Analytical predictions are verified by extensive numerical simulations.
我们提出一种库仑气体方法，用于解析计算稀有事件的概率，其中随机矩阵的最大特征值远大于其典型值。针对威沙特（Wishart）系综和高斯系综，明确计算了表征此概率的大偏差函数。该方法具有通用性，适用于其他相关问题，例如，顶部特征值大幅波动的联合大偏差函数。我们的结果与广泛使用的数据压缩技术即主成分分析相关。通过广泛的数值模拟验证了分析预测。
