How many eigenvalues of a Gaussian random matrix are positive?

Abstract

We study the probability distribution of the index ${\mathcal{N}}_{+}$, i.e., the number of positive eigenvalues of an $N\times N$ Gaussian random matrix. We show analytically that, for large $N$ and large ${\mathcal{N}}_{+}$ with the fraction $0\le c={\mathcal{N}}_{+}/N\le 1$ of positive eigenvalues fixed, the index distribution $\mathcal{P}({\mathcal{N}}_{+}=cN,N)\sim \exp [-\beta N^2\Phi (c)]$ where $\beta$ is the Dyson index characterizing the Gaussian ensemble. The associated large deviation rate function $\Phi (c)$ is computed explicitly for all $0\le c\le 1$. It is independent of $\beta$ and displays a quadratic form modulated by a logarithmic singularity around $c=1/2$. As a consequence, the distribution of the index has a Gaussian form near the peak, but with a variance $\Delta (N)$ of index fluctuations growing as $\Delta (N)\sim \log N/\beta {\pi }^2$ for large $N$. For $\beta =2$, this result is independently confirmed against an exact finite $N$ formula, yielding $\Delta (N)=\log N/2{\pi }^2+C+\mathcal{O}(N^{-1})$ for large $N$, where the constant $C$ has the nontrivial value $C=(\gamma +1+3\log 2)/2{\pi }^2\simeq \mathrm{0.185248...}$ and $\gamma =\mathrm{0.5772...}$ is the Euler constant. We also determine for large $N$ the probability that the interval $[{\zeta }_1,{\zeta }_2]$ is free of eigenvalues. Part of these results have been announced in a recent letter [\textit{Phys. Rev. Lett.} {\bf 103}, 220603 (2009)].

25 pages, 6 figures

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