The Annals of Applied Probability

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

Mihail Poplavskyi, Roger Tribe, and Oleg Zaboronski

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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
Received: April 2016
Revised: July 2016
First available in Project Euclid: 19 July 2017

Permanent link to this document
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

Keywords
Real Ginibre ensemble Fredholm determinant

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


Export citation

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.