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 $X_{s}^{(\max)}\stackrel{D}{=}\sqrt{4s}\lambda_{\max}$.