Pair Dependent Linear Statistics for Circular Beta Ensemble

Abstract

We study limiting distribution of pair counting statistics of the form $ \sum_{1\leq i\neq j\leq N} f(L_N\*(θ_i-θ_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.

Random Matrices: Theory and Applications

References in corpus (3)

• Mesoscopic linear statistics of Wigner matrices
• Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices
• The Berry-Esseen Theorem for Circular $\beta$-ensemble

Cited by in corpus (1)

• A note on pair dependent linear statistics with slowly growing variance