Research output: Contribution to journal › Article

Jinho Baik, Thomas Bothner

Original language | English |
---|---|

Pages (from-to) | 460-501 |

Number of pages | 42 |

Journal | The Annals of Applied Probability |

Volume | 30 |

Issue number | 1 |

DOIs | |

Accepted/In press | 21 Jun 2019 |

Published | 25 Feb 2020 |

Additional links |

**The largest real eigenvalue_BAIK_Accepted_21 Jun 19_GREEN_AAM**The_largest_real_eigenvalue_BAIK_Accepted_21_Jun_19_GREEN_AAM.pdf, 6.44 MB, application/pdf

Uploaded date:27 Feb 2020

Version:Accepted author manuscript

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.

King's College London - Homepage

© 2020 King's College London | Strand | London WC2R 2LS | England | United Kingdom | Tel +44 (0)20 7836 5454