Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
复制标题
DOI：
10.1103/physrevlett.102.060601
复制
发表时间：
2009-02-13
影响因子：
8.6
通讯作者：
Vergassola, Massimo
中科院分区：
文献类型：
--
作者：
Majumdar, Satya N.;Vergassola, Massimo
本文用库仑气体法解析计算了随机矩阵的最大特征值远大于其典型值的罕见事件的概率。表征这种概率的大偏差函数是为Wishart和高斯系综明确计算的。该方法具有通用性，适用于其他相关问题，如顶特征值大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有关，即主成分分析。大量的数值模拟验证了分析预测。
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.