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

We study the large-\(n\) limit of the probability \(p_{2n,2k}\) that a random \(2n\times 2n\) matrix sampled from the real Ginibre ensemble has \(2k\) real eigenvalues. We prove that, $$\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,2k}=\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,0}= -\frac{1}{\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right),$$ where \(\zeta\) is the Riemann zeta-function. Moreover, for any sequence of non-negative integers \((k_n)_{n\geq 1}\), $$\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,2k_n}=-\frac{1}{\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right),$$ provided \(\lim_{n\rightarrow \infty} \left(n^{-1/2}\log(n)\right) k_{n}=0\).