Research output: Contribution to journal › Article

**The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat system.** / Baik, Jinho; Bothner, Thomas.

Research output: Contribution to journal › Article

Baik, J & Bothner, T 2020, 'The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat system', *The Annals of Applied Probability*, vol. 30, no. 1, pp. 460-501. https://doi.org/10.1214/19-AAP1509

Baik, J., & Bothner, T. (2020). The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat system. *The Annals of Applied Probability*, *30*(1), 460-501. https://doi.org/10.1214/19-AAP1509

Baik J, Bothner T. The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat system. The Annals of Applied Probability. 2020 Feb 25;30(1):460-501. https://doi.org/10.1214/19-AAP1509

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title = "The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat system",

abstract = "The real Ginibre ensemble consists of n × n real matrices 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 Rn = max1≤j≤n |zj (X)| of the eigenvalues zj (X) ∈ ℂ of a real Ginibre matrix X follows a different limiting law (as n→∞) for zj (X) ∈ ℝ than for zj (X) ∈ ℂ \ 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 maxj:zj∈ℝ zj (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 maxj:zj∈ℝ zj (X) and extend recent tail estimates in (Ann. Appl. Probab. 27 (2017) 1395.1413) via nonlinear steepest descent techniques.",

keywords = "Deift-Zhou nonlinear steepest descent method, Extreme value statistics, Inverse scattering theory, Real Ginibre ensemble, Riemann-Hilbert problem, Zakharov-Shabat system",

author = "Jinho Baik and Thomas Bothner",

year = "2020",

month = feb,

day = "25",

doi = "10.1214/19-AAP1509",

language = "English",

volume = "30",

pages = "460--501",

journal = "The Annals of Applied Probability",

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T1 - The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat system

AU - Baik, Jinho

AU - Bothner, Thomas

PY - 2020/2/25

Y1 - 2020/2/25

N2 - The real Ginibre ensemble consists of n × n real matrices 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 Rn = max1≤j≤n |zj (X)| of the eigenvalues zj (X) ∈ ℂ of a real Ginibre matrix X follows a different limiting law (as n→∞) for zj (X) ∈ ℝ than for zj (X) ∈ ℂ \ 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 maxj:zj∈ℝ zj (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 maxj:zj∈ℝ zj (X) and extend recent tail estimates in (Ann. Appl. Probab. 27 (2017) 1395.1413) via nonlinear steepest descent techniques.

AB - The real Ginibre ensemble consists of n × n real matrices 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 Rn = max1≤j≤n |zj (X)| of the eigenvalues zj (X) ∈ ℂ of a real Ginibre matrix X follows a different limiting law (as n→∞) for zj (X) ∈ ℝ than for zj (X) ∈ ℂ \ 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 maxj:zj∈ℝ zj (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 maxj:zj∈ℝ zj (X) and extend recent tail estimates in (Ann. Appl. Probab. 27 (2017) 1395.1413) via nonlinear steepest descent techniques.

KW - Deift-Zhou nonlinear steepest descent method

KW - Extreme value statistics

KW - Inverse scattering theory

KW - Real Ginibre ensemble

KW - Riemann-Hilbert problem

KW - Zakharov-Shabat system

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U2 - 10.1214/19-AAP1509

DO - 10.1214/19-AAP1509

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VL - 30

SP - 460

EP - 501

JO - The Annals of Applied Probability

JF - The Annals of Applied Probability

SN - 1050-5164

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