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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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