In this paper, we consider the maximum of the Sine β counting process from its expectation. We show the leading order behavior is consistent with the predictions of log-correlated Gaussian fields, also consistent with work on the imaginary part of the log-characteristic polynomial of random matrices. We do this by a direct analysis of the stochastic sine equation, which gives a description of the continuum limit of the Prüfer phases of a Gaussian β-ensemble matrix.
We study the scaling limit of the spectrum of the β-Jacobi ensemble at the soft-edge and hard-edge for general values of β. We show that the limiting point processes correspond respectively to the stochastic Airy and Bessel point processes introduced in [19] and [20].
We give overcrowding estimates for the Sine β process, the bulk point process limit of the Gaussian β-ensemble. We show that the probability of having at least n points in a fixed interval is given by e − β 2 n 2 log(n)+O(n 2 ) as n → ∞. We also identify the next order term in the exponent if the size of the interval goes to zero.
We study properties of the point process that appears as the local limit at the random matrix hard edge. We show a transition from the hard edge to bulk behavior and give a central limit theorem and large deviation result for the number of points in a growing interval [0, λ] as λ → ∞. We study these results for the square root of the hard edge process. In this setting many of these behaviors mimic those of the Sine β process.
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