ON THE DISTRIBUTION OF THE LARGEST REAL EIGENVALUE FOR THE REAL GINIBRE ENSEMBLE

• 成果类型: Article
• 署名作者: Poplavskyi, Mihail; Tribe, Roger; Zaboronski, Oleg
• 署名单位: University of Warwick
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1233
• 发表日期: 2017
• 页码: 1395-1413
• 关键词: random matrices; UNIVERSALITY; spectrum; edge

摘要

Let root N + lambda(maX), be the largest real eigenvalue of a random N x N matrix with independent N(0, 1) entries (the real Ginibre matrix). We study the large deviations behaviour of the limiting N -> infinity 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 - 1/4 erfc(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): for t < 0, P[lambda(max) < t] = e(1/2 root 2 pi zeta(3/2)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; see [Garrod, Poplayskyi, Tribe and Zaboronski (2015)]. 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)) (D) double under bar root 4s lambda(max).