Open Access
October 2016 What is the probability that a large random matrix has no real eigenvalues?
Eugene Kanzieper, Mihail Poplavskyi, Carsten Timm, Roger Tribe, Oleg Zaboronski
Ann. Appl. Probab. 26(5): 2733-2753 (October 2016). DOI: 10.1214/15-AAP1160


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 $\lim_{n\rightarrow\infty}(n^{-1/2}\log(n))k_{n}=0$.


Download Citation

Eugene Kanzieper. Mihail Poplavskyi. Carsten Timm. Roger Tribe. Oleg Zaboronski. "What is the probability that a large random matrix has no real eigenvalues?." Ann. Appl. Probab. 26 (5) 2733 - 2753, October 2016.


Received: 1 April 2015; Published: October 2016
First available in Project Euclid: 19 October 2016

zbMATH: 1375.60019
MathSciNet: MR3563192
Digital Object Identifier: 10.1214/15-AAP1160

Primary: 60B20
Secondary: 60F10

Keywords: large deviations , Real Ginibre ensemble

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.26 • No. 5 • October 2016
Back to Top