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

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 $P[\lambda_{max}<t]$ of the shifted maximal real eigenvalue $\lambda_{max}$. In particular, we prove that the right tail of this distribution is Gaussian: for $t>0$, \[ P[\lambda_{max}<t]=1-\frac{1}{4}\mbox{erfc}(t)+O\left(e^{-2t^2}\right). \] This is a rigorous confirmation of the corresponding result of Forrester and Nagao. We also prove that the left tail is exponential: for $t<0$, \[ P[\lambda_{max}<t]= e^{\frac{1}{2\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right)t+O(1)}, \] 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 (ABM's) with the step initial condition. 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 $X_s^{(max)}\stackrel{D}{=} \sqrt{4s}\lambda_{max}$.