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

Let $\sqrt{\mathrm{N}}+{\mathrm{\lambda }}_{\mathrm{max}}$ be the largest real eigenvalue of a random N . N matrix with independent N(0, 1) entries (the "real Ginibre matrix"). We study the large deviations behaviour of the limiting N . . distribution .[.max < t] of the shifted maximal real eigenvalue .max. In particular, we prove that the right tail of this distribution is Gaussian: for t > 0, $\mathrm{\mathbb{P}}[{\mathrm{\lambda }}_{\mathrm{max}}<\mathrm{t}]=1-\frac{1}{4}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\left(\mathrm{t}\right)+\mathrm{O}\left({\mathrm{e}}^{-2{\mathrm{t}}^{2}}\right)$. 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, $\mathrm{\mathbb{P}}[{\mathrm{\lambda }}_{\mathrm{max}}<\mathrm{t}]={\mathrm{e}}^{\frac{1}{2\sqrt{2\mathrm{\pi }}}\mathrm{\zeta }\left(\frac{3}{2}\right)\mathrm{t}+\mathrm{O}\left(1\right)}$, where . 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 ${\mathrm{X}}_{\mathrm{s}}^{(\mathrm{max})}$.the position of the rightmost annihilating particle at fixed time s > 0.can be read off from the corresponding answers for .max using ${\mathrm{X}}_{\mathrm{s}}^{(\mathrm{max})}\stackrel{\mathrm{D}}{=}\sqrt{4\mathrm{s}}{\mathrm{\lambda }}_{\mathrm{max}}$.