The Annals of Probability

Extremal eigenvalue correlations in the GUE minor process and a law of fractional logarithm

Elliot Paquette and Ofer Zeitouni

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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, $c_{1}=c_{2}=4^{1/3}$.

Article information

Ann. Probab., Volume 45, Number 6A (2017), 4112-4166.

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

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Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F99: None of the above, but in this section

GUE law of fractional logarithm minor process


Paquette, Elliot; Zeitouni, Ofer. Extremal eigenvalue correlations in the GUE minor process and a law of fractional logarithm. Ann. Probab. 45 (2017), no. 6A, 4112--4166. doi:10.1214/16-AOP1161.

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