We compute analytically, for large $N$, the probability distribution of the number of positive eigenvalues (the index $N_{+}$) of a random $N\times N$ 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 $\mathcal{P}(c,N)\simeq \exp [-\beta N^2\Phi (c)]$ 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.

• Received 5 October 2009

DOI:https://doi.org/10.1103/PhysRevLett.103.220603