Gaussian Fluctuation for Smoothed Local Correlations in CUE

Abstract

AbstractMotivated by the Rudnick-Sarnak theorem we study limiting distribution of smoothed local correlations of the form $$\begin{aligned} \sum _{j_1, j_2, \ldots , j_n} f(N(\theta _{j_2}-\theta _{j_1}), N(\theta _{j_3}-\theta _{j_1}), \ldots , N(\theta _{j_n}-\theta _{j_1})) \end{aligned}$$ ∑ j 1 , j 2 , … , j n f ( N ( θ j 2 - θ j 1 ) , N ( θ j 3 - θ j 1 ) , … , N ( θ j n - θ j 1 ) ) for the Circular United Ensemble of random matrices for sufficiently smooth test functions.