## The Annals of Applied Probability

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

#### Article information

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

Dates
First available in Project Euclid: 19 October 2016

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

#### 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

#### References

• [1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. U.S. Government Printing Office, Washington, D.C.
• [2] Akemann, G. and Kanzieper, E. (2007). Integrable structure of Ginibre’s ensemble of real random matrices and a Pfaffian integration theorem. J. Stat. Phys. 129 1159–1231.
• [3] Ben Arous, G. and Zeitouni, O. (1998). Large deviations from the circular law. ESAIM Probab. Stat. 2 123–134 (electronic).
• [4] Bleher, P. and Mallison, R. Jr. (2006). Zeros of sections of exponential sums. Int. Math. Res. Not. IMRN Art. ID 38937, 49.
• [5] Borodin, A. and Kanzieper, E. (2007). A note on the Pfaffian integration theorem. J. Phys. A 40 F849–F855.
• [6] Borodin, A. and Sinclair, C. D. (2009). The Ginibre ensemble of real random matrices and its scaling limits. Comm. Math. Phys. 291 177–224.
• [7] del Molino, L. C. G., Pakdaman, K., Touboul, J. and Wainrib, G. (2015). The Ginibre ensemble with $k=O(n)$ real eigenvalues. Available at arXiv:1501.03120v1.
• [8] Derrida, B. and Zeitak, R. (1996). Distribution of domain sizes in the zero temperature Glauber dynamics of the one-dimensional Potts model. Phys. Rev. E 54 2513–2525.
• [9] Edelman, A. (1997). The probability that a random real Gaussian matrix has $k$ real eigenvalues, related distributions, and the circular law. J. Multivariate Anal. 60 203–232.
• [10] Edelman, A., Kostlan, E. and Shub, M. (1994). How many eigenvalues of a random matrix are real? J. Amer. Math. Soc. 7 247–267.
• [11] Forrester, P. J. (2015). Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble. J. Phys. A 48 324001, 14.
• [12] Forrester, P. J. and Nagao, T. (2007). Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99 050603.
• [13] Ginibre, J. (1965). Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6 440–449.
• [14] Kanzieper, E. and Akemann, G. (2005). Statistics of real eigenvalues in Ginibre’s ensemble of random real matrices. Phys. Rev. Lett. 95 230201, 4.
• [15] Macdonald, I. G. (1995). Symmetric Functions and Hall Polynomials, 2nd ed. Clarendon Press, Oxford.
• [16] Tribe, R. and Zaboronski, O. (2011). Pfaffian formulae for one dimensional coalescing and annihilating systems. Electron. J. Probab. 16 2080–2103.