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
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