Global Fluctuations for Linear Statistics of \beta-Jacobi Ensembles

TL;DR

This paper investigates the asymptotic Gaussian fluctuations of linear eigenvalue statistics in eta-Jacobi ensembles, providing explicit covariance structures and a law of large numbers for polynomial functions.

Contribution

It introduces a detailed analysis of fluctuations for eta-Jacobi ensembles, including covariance computation and basis diagonalization, extending understanding of eigenvalue behavior.

Findings

Fluctuations are asymptotically Gaussian.

Covariance matrix is diagonalized by shifted Chebyshev polynomials.

A law of large numbers is established for polynomial test functions.

Abstract

We study the global fluctuations for linear statistics of the form $\sum_{i=1}^n f(\lambda_i)$ as $n \rightarrow \infty$, for $C^1$ functions $f$, and $\lambda_1, ..., \lambda_n$ being the eigenvalues of a (general) $\beta$-Jacobi ensemble, for which tridiagonal models were given by Killip and Nenciu as well as Edelman and Sutton. The fluctuation from the mean ($\sum_{i=1}^n f(\lambda_i) - \Exp \sum_{i=1}^n f(\lambda_i)$) is given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation from the predicted mean for polynomial test functions, and we obtain a law of large numbers.