The Index Distribution of Gaussian Random Matrices

Abstract

We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N_{+}) of a random NxN matrix belonging to Gaussian orthogonal (β=1), unitary (β=2) or symplectic (β=4) ensembles. The distribution of the fraction of positive eigenvalues c=N_{+}/N scales, for large N, as Prob(c,N)\simeq\exp[-βN^2 Φ(c)] where the rate function Φ(c), symmetric around c=1/2 and universal (independent of $β$), is calculated exactly. The distribution has non-Gaussian tails, but even near its peak at c=1/2 it is not strictly Gaussian due to an unusual logarithmic singularity in the rate function.