How many eigenvalues of a Gaussian random matrix are positive?

TL;DR

This paper analytically characterizes the probability distribution of the number of positive eigenvalues in large Gaussian random matrices, revealing a universal large deviation form and confirming results with exact finite-size formulas.

Contribution

It provides an explicit large deviation rate function for the index distribution of Gaussian matrices, independent of Dyson index, and confirms the variance growth with exact finite-size analysis.

Findings

Large deviation form of index distribution derived

Variance of positive eigenvalues grows as log N / (βπ²)

Exact finite N formula confirms asymptotic variance result

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 $0\leq c \leq 1$. 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 $C=(\gamma+1+3\log 2)/2\pi^2\simeq 0.185248...$ and $\gamma=0.5772...$ 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)].