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 $Ntimes N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix). We study the large deviations behaviour of the limiting $Nrightarrow infty$ 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)+Oleft(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}{2sqrt{2pi}}zetaleft(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 (ABMs) 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 $X_s^{(max)}stackrel{D}{=} sqrt{4s}lambda_{max}$.