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

Abstract

Let N−−√+λmaxN+λmax be the largest real eigenvalue of a random N×NN×N matrix with independent N(0,1)N(0,1) entries (the “real Ginibre matrix”). We study the large deviations behaviour of the limiting N→∞N→∞ distribution P[λmax 0t>0, P[λmax<t]=1−14erfc(t)+O(e−2t2). P[λmax<t]=1−14erfc⁡(t)+O(e−2t2). 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)O(1): for t<0t<0, P[λmax<t]=e122π√ζ(32)t+O(1), P[λmax<t]=e122πζ(32)t+O(1), where ζζ 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(max)sXs(max)—the position of the rightmost annihilating particle at fixed time s>0s>0—can be read off from the corresponding answers for λmaxλmax using X(max)s=D4s−−√λmaxXs(max)=D4sλmax.