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

TL;DR

This paper develops a Coulomb gas method to analytically compute the probability of rare large deviations of the maximum eigenvalue in Wishart and Gaussian random matrices, with implications for data analysis techniques.

Contribution

Introduces a simple Coulomb gas approach to derive explicit large deviation functions for maximum eigenvalues in Wishart and Gaussian matrices, applicable to related problems.

Findings

Explicit large deviation functions derived for Wishart and Gaussian ensembles.

Method verified by extensive numerical simulations.

Results relevant to principal components analysis and data compression.

Abstract

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.