## Extremal eigenvalue correlations in the GUE minor process and a law of fractional logarithm.(English)Zbl 1390.60036

Let $$(Z_{i,j})_{i,j=1}^\infty$$ be an infinite $$\text{GUE}$$ matrix, that is a doubly-infinite array of random variables such that $$Z_{i,j}$$ is a centered complex Gaussian random variable of variance $$1/2$$ for all $$i > j$$, $$Z_{i,i}$$ is a centered real Gaussian variable of variance $$1/2$$ for all $$i$$, all these variables are mutually independent, and $$Z_{j,i} = \overline{Z_{i,j}}$$. Let $$\tilde \lambda_N$$ be the largest eigenvalue of the standard $$\text{GUE}(N)$$-matrix $$(Z_{i,j})_{i,j=1}^N$$. Center and scale it by defining $\lambda_N := \frac{\tilde \lambda_N - \sqrt{2N}}{\sqrt 2 N^{1/6}}.$ Then, it is known that $$\lambda_N$$ converges in distribution to a Tracy-Widom random variable. The authors prove that $\limsup_{N\to\infty} \frac{\lambda_N}{(\log N)^{2/3}} = 4^{-2/3}$ almost surely, and that for some constants $$c_1,c_2>0$$, $-c_1 \leq \liminf_{N\to\infty} \frac{\lambda_N}{(\log N)^{1/3}} = 4^{-2/3}\leq -c_2$ with probability $$1$$. The authors conjecture that in fact, $$c_1=c_2 = 4^{1/3}$$.

### MSC:

 60B20 Random matrices (probabilistic aspects) 60F15 Strong limit theorems
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