Open Access
June 2017 On the distribution of the largest real eigenvalue for the real Ginibre ensemble
Mihail Poplavskyi, Roger Tribe, Oleg Zaboronski
Ann. Appl. Probab. 27(3): 1395-1413 (June 2017). DOI: 10.1214/16-AAP1233

Abstract

Let N+λmax 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, P[λmax<t]=114erfc(t)+O(e2t2). 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[λmax<t]=e122πζ(32)t+O(1), where ζ 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 Xs(max)—the position of the rightmost annihilating particle at fixed time s>0—can be read off from the corresponding answers for λmax using Xs(max)=D4sλmax.

Citation

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Mihail Poplavskyi. Roger Tribe. Oleg Zaboronski. "On the distribution of the largest real eigenvalue for the real Ginibre ensemble." Ann. Appl. Probab. 27 (3) 1395 - 1413, June 2017. https://doi.org/10.1214/16-AAP1233

Information

Received: 1 April 2016; Revised: 1 July 2016; Published: June 2017
First available in Project Euclid: 19 July 2017

zbMATH: 1375.60023
MathSciNet: MR3678474
Digital Object Identifier: 10.1214/16-AAP1233

Subjects:
Primary: 60B20
Secondary: 60F10

Keywords: Fredholm determinant , Real Ginibre ensemble

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.27 • No. 3 • June 2017
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