The Index Distribution of Gaussian Random Matrices

TL;DR

This paper analytically derives the large N probability distribution of the number of positive eigenvalues in Gaussian random matrices from different ensembles, revealing a universal, non-Gaussian distribution with unique singularities.

Contribution

It provides an exact calculation of the rate function for the distribution of positive eigenvalues in Gaussian ensembles, highlighting universality and non-Gaussian features.

Findings

Distribution scales as exp(-β N^2 Φ(c)) for large N

Rate function Φ(c) is universal and symmetric around c=1/2

Distribution exhibits non-Gaussian tails and logarithmic singularities

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 (\beta=1), unitary (\beta=2) or symplectic (\beta=4) ensembles. The distribution of the fraction of positive eigenvalues c=N_{+}/N scales, for large N, as Prob(c,N)\simeq\exp[-\beta N^2 \Phi(c)] where the rate function \Phi(c), symmetric around c=1/2 and universal (independent of $\beta$), 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.