The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles

TL;DR

This paper derives an explicit formula for the average spectral measure of a class of random Jacobi matrices, which are limits of Gaussian beta ensemble matrices under specific scaling conditions.

Contribution

It provides a new explicit formula for the mean spectral measure of these matrices, connecting random matrix theory with spectral analysis.

Findings

Explicit formula for the mean spectral measure of the matrices.

Connection established between Jacobi matrices and Gaussian beta ensembles.

Results applicable in the limit as matrix size grows with fixed Nβ.

Abstract

An explicit formula for the mean spectral measure of a random Jacobi matrix is derived. The matrix may be regarded as the limit of Gaussian beta ensemble (G$\beta$E) matrices as the matrix size $N$ tends to infinity with the constraint that $N \beta $ is a constant.