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February 2020 The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat system
Jinho Baik, Thomas Bothner
Ann. Appl. Probab. 30(1): 460-501 (February 2020). DOI: 10.1214/19-AAP1509

Abstract

The real Ginibre ensemble consists of $n\times n$ real matrices $\mathbf{X}$ whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius $R_{n}=\max_{1\leq j\leq n}|z_{j}(\mathbf{X})|$ of the eigenvalues $z_{j}(\mathbf{X})\in \mathbb{C}$ of a real Ginibre matrix $\mathbf{X}$ follows a different limiting law (as $n\rightarrow \infty $) for $z_{j}(\mathbf{X})\in \mathbb{R}$ than for $z_{j}(\mathbf{X})\in \mathbb{C}\setminus \mathbb{R}$. Building on previous work by Rider and Sinclair (Ann. Appl. Probab. 24 (2014) 1621–1651) and Poplavskyi, Tribe and Zaboronski (Ann. Appl. Probab. 27 (2017) 1395–1413), we show that the limiting distribution of $\max_{j:z_{j}\in \mathbb{R}}z_{j}(\mathbf{X})$ admits a closed-form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov–Shabat system. As byproducts of our analysis, we also obtain a new determinantal representation for the limiting distribution of $\max_{j:z_{j}\in \mathbb{R}}z_{j}(\mathbf{X})$ and extend recent tail estimates in (Ann. Appl. Probab. 27 (2017) 1395–1413) via nonlinear steepest descent techniques.

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Jinho Baik. Thomas Bothner. "The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat system." Ann. Appl. Probab. 30 (1) 460 - 501, February 2020. https://doi.org/10.1214/19-AAP1509

Information

Received: 1 August 2018; Revised: 1 June 2019; Published: February 2020
First available in Project Euclid: 25 February 2020

zbMATH: 07200533
MathSciNet: MR4068316
Digital Object Identifier: 10.1214/19-AAP1509

Subjects:
Primary: 60B20
Secondary: 45M05, 60G70

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 1 • February 2020
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