## The Annals of Applied Probability

### On the distribution of the largest real eigenvalue for the real Ginibre ensemble

#### Abstract

Let $\sqrt{N}+\lambda_{\max}$ be the largest real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the “real Ginibre matrix”). We study the large deviations behaviour of the limiting $N\rightarrow \infty$ distribution $\mathbb{P}[\lambda_{\max}<t]$ of the shifted maximal real eigenvalue $\lambda_{\max}$. In particular, we prove that the right tail of this distribution is Gaussian: for $t>0$, \begin{equation*}\mathbb{P}[\lambda_{\max}<t]=1-\frac{1}{4}\operatorname{erfc}(t)+O(e^{-2t^{2}}).\end{equation*} This is a rigorous confirmation of the corresponding result of [Phys. Rev. Lett. 99 (2007) 050603]. We also prove that the left tail is exponential, with correct asymptotics up to $O(1)$: for $t<0$, \begin{equation*}\mathbb{P}[\lambda_{\max}<t]=e^{\frac{1}{2\sqrt{2\pi }}\zeta (\frac{3}{2})t+O(1)},\end{equation*} where $\zeta$ is the Riemann zeta-function.

Our results have implications for interacting particle systems. The edge scaling limit of the law of real eigenvalues for the real Ginibre ensemble is a rescaling of a fixed time distribution of annihilating Brownian motions (ABMs) with the step initial condition; see [Garrod, Poplavskyi, Tribe and Zaboronski (2015)]. Therefore, the tail behaviour of the distribution of $X_{s}^{(\max)}$—the position of the rightmost annihilating particle at fixed time $s>0$—can be read off from the corresponding answers for $\lambda_{\max}$ using $X_{s}^{(\max)}\stackrel{D}{=}\sqrt{4s}\lambda_{\max}$.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 3 (2017), 1395-1413.

Dates
Revised: July 2016
First available in Project Euclid: 19 July 2017

https://projecteuclid.org/euclid.aoap/1500451226

Digital Object Identifier
doi:10.1214/16-AAP1233

Mathematical Reviews number (MathSciNet)
MR3678474

Zentralblatt MATH identifier
1375.60023

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

#### Citation

Poplavskyi, Mihail; Tribe, Roger; Zaboronski, Oleg. On the distribution of the largest real eigenvalue for the real Ginibre ensemble. Ann. Appl. Probab. 27 (2017), no. 3, 1395--1413. doi:10.1214/16-AAP1233. https://projecteuclid.org/euclid.aoap/1500451226

#### References

• [1] Borodin, A. and Gorin, V. (2012). Lectures on integrable probability. Preprint. Available at arXiv:1212.3351v2.
• [2] Borodin, A. and Sinclair, C. D. (2009). The Ginibre ensemble of real random matrices and its scaling limits. Comm. Math. Phys. 291 177–224.
• [3] Deift, P. and Gioev, D. (2007). Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices. Comm. Pure Appl. Math. 60 867–910.
• [4] Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. and Zhou, X. (1999). Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 1335–1425.
• [5] 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.
• [6] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
• [7] Forrester, P. J. (2015). Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble. J. Phys. A 48 324001.
• [8] Forrester, P. J. and Nagao, T. (2007). Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99 050603.
• [9] Garrod, B., Poplavskyi, M., Tribe, R. and Zaboronski, O. (2015). Interacting particle systems on $\mathbf{Z}$ as Pfaffian point processes I—Annihilating and coalescing random walks. Preprint. Available at arXiv:1507.01843.
• [10] Ginibre, J. (1965). Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6 440–449.
• [11] Krapivsky, P. L. and Ben-Naim, E. (2015). Irreversible reactions and diffusive escape: Stationary properties. J. Stat. Mech. Theory Exp. P05003.
• [12] Krapivsky, P. L., Mallick, K. and Sadhu, T. (2015). Melting of an Ising quadrant. J. Phys. A 48 015005.
• [13] Lotov, V. I. (1996). On some boundary crossing problems for Gaussian random walks. Ann. Probab. 24 2154–2171.
• [14] Rider, B. and Sinclair, C. D. (2014). Extremal laws for the real Ginibre ensemble. Ann. Appl. Probab. 24 1621–1651.
• [15] Sommers, H.-J. (2007). Symplectic structure of the real Ginibre ensemble. J. Phys. A 40 F671–F676.
• [16] Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 697–733.
• [17] Tao, T. and Vu, V. (2015). Random matrices: Universality of local spectral statistics of non-Hermitian matrices. Ann. Probab. 43 782–874.
• [18] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.
• [19] Tracy, C. A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727–754.
• [20] Tribe, R. and Zaboronski, O. (2011). Pfaffian formulae for one dimensional coalescing and annihilating systems. Electron. J. Probab. 16 2080–2103.