What is the probability that a large random matrix has no real eigenvalues?

Abstract

We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2n\times2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that

\[\lim_{n\rightarrow\infty}\frac{1}{\sqrt{2n}}\log p_{2n,2k}=\lim_{n\rightarrow\infty}\frac{1}{\sqrt{2n}}\log p_{2n,0}=-\frac{1}{\sqrt{2\pi}}\zeta (\frac{3}{2}),\] where $\zeta$ is the Riemann zeta-function. Moreover, for any sequence of nonnegative integers $(k_{n})_{n\geq1}$,

\[\lim_{n\rightarrow\infty}\frac{1}{\sqrt{2n}}\log p_{2n,2k_{n}}=-\frac{1}{\sqrt{2\pi}}\zeta (\frac{3}{2}),\] provided $\lim_{n\rightarrow\infty}(n^{-1/2}\log(n))k_{n}=0$.