The Annals of Applied Probability

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

Eugene Kanzieper, Mihail Poplavskyi, Carsten Timm, Roger Tribe, and Oleg Zaboronski

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

Article information

Source
Ann. Appl. Probab., Volume 26, Number 5 (2016), 2733-2753.

Dates
Received: April 2015
First available in Project Euclid: 19 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1476884302

Digital Object Identifier
doi:10.1214/15-AAP1160

Mathematical Reviews number (MathSciNet)
MR3563192

Zentralblatt MATH identifier
1375.60019

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 60F10: Large deviations

Keywords
Real Ginibre ensemble large deviations

Citation

Kanzieper, Eugene; Poplavskyi, Mihail; Timm, Carsten; Tribe, Roger; Zaboronski, Oleg. What is the probability that a large random matrix has no real eigenvalues?. Ann. Appl. Probab. 26 (2016), no. 5, 2733--2753. doi:10.1214/15-AAP1160. https://projecteuclid.org/euclid.aoap/1476884302


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