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Pair Dependent Linear Statistics for Circular Beta Ensemble (1912.07110v3)
Published 15 Dec 2019 in math.PR, math-ph, and math.MP
Abstract: We study limiting distribution of pair counting statistics of the form $ \sum_{1\leq i\neq j\leq N} f(L_N*(\theta_i-\theta_j))$ for the circular $\beta$-ensemble (C$\beta$E) of random matrices for sufficiently smooth test function $f$ and $L_N=O(N).$ For $\beta=2$ and $L_N=N$ our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.
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