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November 2017 Extremal eigenvalue correlations in the GUE minor process and a law of fractional logarithm
Elliot Paquette, Ofer Zeitouni
Ann. Probab. 45(6A): 4112-4166 (November 2017). DOI: 10.1214/16-AOP1161

Abstract

Let $\lambda^{(N)}$ be the largest eigenvalue of the $N\times N$ GUE matrix which is the $N$th element of the GUE minor process, rescaled to converge to the standard Tracy–Widom distribution. We consider the sequence $\{\lambda^{(N)}\}_{N\geq1}$ and prove a law of fractional logarithm for the $\limsup$:

\[\limsup_{N\to\infty}\frac{\lambda^{({{N}})}}{(\log N)^{2/3}}=(\frac{1}{4})^{2/3}\qquad \mbox{almost surely}.\] For the $\liminf$, we prove the weaker result that there are constants $c_{1},c_{2}>0$ so that

\[-c_{1}\leq\liminf_{N\to\infty}\frac{\lambda^{({{N}})}}{(\log N)^{1/3}}\leq-c_{2}\qquad \mbox{almost surely}.\] We conjecture that in fact, $c_{1}=c_{2}=4^{1/3}$.

Citation

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Elliot Paquette. Ofer Zeitouni. "Extremal eigenvalue correlations in the GUE minor process and a law of fractional logarithm." Ann. Probab. 45 (6A) 4112 - 4166, November 2017. https://doi.org/10.1214/16-AOP1161

Information

Received: 1 June 2015; Revised: 1 October 2016; Published: November 2017
First available in Project Euclid: 27 November 2017

zbMATH: 06838117
MathSciNet: MR3729625
Digital Object Identifier: 10.1214/16-AOP1161

Subjects:
Primary: 60B20 , 60F99

Keywords: GUE , law of fractional logarithm , Minor process

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 6A • November 2017
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