Pair Dependent Linear Statistics for Circular Beta Ensemble
Abstract
We study limiting distribution of pair counting statistics of the form Σ1≤ i≠ j≤ N f(LN\*(θi-θj)) for the circular β-ensemble (CβE) of random matrices for sufficiently smooth test function f and LN=O(N). For β=2 and LN=N our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.
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