Paper detail

How many eigenvalues of a Gaussian random matrix are positive?

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 $\mathcal{P}({\mathcal N}_+=cN,N)\sim\exp[-βN^2 Φ(c)]$ where $β$ is the Dyson index characterizing the Gaussian ensemble. The associated large deviation rate function $Φ(c)$ is computed explicitly for all $0\leq c \leq 1$. It is independent of $β$ 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 $Δ(N)$ of index fluctuations growing as $Δ(N)\sim \log N/βπ^2$ for large $N$. For $β=2$, this result is independently confirmed against an exact finite $N$ formula, yielding $Δ(N)= \log N/2π^2 +C+\mathcal{O}(N^{-1})$ for large $N$, where the constant $C$ has the nontrivial value $C=(γ+1+3\log 2)/2π^2\simeq 0.185248...$ and $γ=0.5772...$ is the Euler constant. We also determine for large $N$ the probability that the interval $[ζ_1,ζ_2]$ is free of eigenvalues. Part of these results have been announced in a recent letter [\textit{Phys. Rev. Lett.} {\bf 103}, 220603 (2009)].