Since their introduction in 1962 by Freeman Dyson, the Circular-Beta Random matrix ensembles (\(C\beta E\)) have been extensively studied due to their underlying connection to statistics that appear in various areas of mathematics and physics. It was shown by \(H\). Montgomery (1973) that, assuming the Riemann Hypothesis, the pair correlation statistics for the rescaled zeros of the Riemann Zeta function on the critical line are the same as those for the rescaled eigenvalues of large unitary matrices distributed according to Haar measure (\(C\beta E\) with \(\beta=2\)). The results in this talk, many of which are based on joint work with A. Aguirre and A. Soshnikov, are motivated by the work of Montgomery.

First, let \(\theta_1...\theta_N\) be the angles of the eigenvalues of an \(NxN\) unitary matrix sampled from the collection of \(NxN\) unitary matrices, \(U(N)\), according to Haar measure. We consider the asymptotic distribution of statistics of the form \(S_N(f)=\frac{1}{\sqrt{L_N}}\sum_{1\leq i\neq j\leq N} f(L_N(\theta_i-\theta_j))\), where \(f\) is suitable test function and \(L_N\) is a positive, non-decreasing sequence satisfying \(L_N/N \to 0\) as \(N \to \infty\). We prove a variety of asymptotic results regarding the limiting distribution of \(S_N(f)\) as \(N \to \infty\). In the case where \(L_N=1\) (global statistics), the limiting distribution is an infinite sum of exponential random variables. When \(1<<L_N\leq N\), the limiting distribution is normal, i.e. we obtain central limit theorems. Second, we consider the same type of linear statistics, but where \(\theta_1...\theta_N\) are distributed according to Haar measure on two other classical compact matrix groups: \(SO(N)\) and \(Sp(N)\). In this case, when \(L_N=1\), the limiting distribution is an infinite sum of non-central chi squared random variables. When \(1<<L_N<< N\), we again prove central theorems similar to those for the unitary matrices.