Accepted Friday Nov 06, 2009
We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N+) of a random N×N matrix belonging to Gaussian orthogonal (b = 1), unitary (b = 2) or symplectic (b = 4) ensembles. The distribution of the fraction of positive eigenvalues c=N+/N scales, for large N, as P(c,N) @ exp[-bN2 F(c)] where the rate function F(c), symmetric around c=1/2 and universal (independent of b), 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.