Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices

Details Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices

2008-11-14

arxiv.org/abs/0811.2290v1

Phys.Rev.Lett.102:060601,2009

Physics

Condensed Matter

Statistical Mechanics

4 pages, 3 .eps figures included

Scientific paper

10.1103/PhysRevLett.102.060601

We present a simple 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 quite 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.

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