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 $2n\times 2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that, $$\underset{n\to \infty }{\lim }\frac{1}{\sqrt{2n}}\log p_{2n,2k}=\underset{n\to \infty }{\lim }\frac{1}{\sqrt{2n}}\log p_{2n,0}=-\frac{1}{\sqrt{2\pi }}\zeta \left(\frac{3}{2}\right),$$ where $\zeta$ is the Riemann zeta-function. Moreover, for any sequence of non-negative integers $(k_n{)}_{n\ge 1}$, $$\underset{n\to \infty }{\lim }\frac{1}{\sqrt{2n}}\log p_{2n,2k_n}=-\frac{1}{\sqrt{2\pi }}\zeta \left(\frac{3}{2}\right),$$ provided ${\lim }_{n\to \infty }\left(n^{-1/2}\log (n)\right)k_n=0$.

23 pages, 1 figure

References in corpus (2)

• Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices
• A Note on the Pfaffian Integration Theorem

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• The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble