On spectral measures of random Jacobi matrices

TL;DR

This paper investigates the asymptotic spectral behavior of random Jacobi matrices from Gaussian, Wishart, and MANOVA beta ensembles, revealing convergence to classical distributions and analyzing Gaussian fluctuations.

Contribution

It establishes the weak convergence of spectral measures to known distributions and explores Gaussian fluctuations around these limits for the first time.

Findings

Spectral measures converge to semicircle, Marchenko-Pastur, and Kesten-McKay distributions.

Gaussian fluctuations around the limits are characterized.

Results apply to Gaussian, Wishart, and MANOVA beta ensembles.

Abstract

The paper studies the limiting behavior of spectral measures of random Jacobi matrices of Gaussian, Wishart and MANOVA beta ensembles. We show that the spectral measures converge weakly to a limit distribution which is the semicircle distribution, Marchenko-Pastur distributions or Kesten-Mckey distributions, respectively. The Gaussian fluctuation around the limit is then investigated.