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 ℙ[λmax < t] of the shifted maximal real eigenvalue λmax. In particular, we prove that the right tail of this distribution is Gaussian: for t > 0, $\mathrm{\mathbb{P}}[{\mathrm{\lambda }}_{\mathrm{max}}