## The Annals of Applied Probability

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

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

Mihail Poplavskyi, Roger Tribe, and Oleg Zaboronski

#### 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