Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices

Abstract

We compute analytically the probability of large fluctuations to the left of the mean of the largest eigenvalue in the Wishart (Laguerre) ensemble of positive definite random matrices. We show that the probability that all the eigenvalues of a (N x N) Wishart matrix W=X^T X (where X is a rectangular M x N matrix with independent Gaussian entries) are smaller than the mean value <λ>=N/c decreases for large N as $\sim \exp[-\fracβ{2}N^2 Φ_{-}(\frac{2}{\sqrt{c}}+1;c)]$, where β=1,2 correspond respectively to real and complex Wishart matrices, c=N/M < 1 and Φ_{-}(x;c) is a large deviation function that we compute explicitly. The result for the Anti-Wishart case (M < N) simply follows by exchanging M and N. We also analytically determine the average spectral density of an ensemble of constrained Wishart matrices whose eigenvalues are forced to be smaller than a fixed barrier. The numerical simulations are in excellent agreement with the analytical predictions.