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Pair Dependent Linear Statistics for Circular Random Matrix Ensembles


Let $\theta_1,...,\theta_N$ be the angles of the eigenvalues of a $N\times N$ matrix sampled from either $C\beta E$, $SO(N)$, or $Sp(N)$. In this dissertation, we study the limiting distribution of the "pair dependent" linear statistic $\left(\frac{1}{\sqrt{L_N}}\right)\sum_{1\leq i\neq j\leq N} f(L_N(\theta_i-\theta_j))$, where $f$ is a sufficiently smooth function and $L_N$ is a positive, non-decreasing sequence such that $1\leq L_N<< N$. When $L_N=1$ (global case), the limiting distribution is an infinite sum of independent random variables, exponential in the case of $C\beta E$ and chi-squared distributed in the cases of $SO(N)$ and $Sp(N)$. When $L_N\to \infty$ (mesoscopic case), we are able to prove central limit theorems for each of the mentioned random matrix ensembles.

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