Spectral Properties of the Jacobi Ensembles via the Coulomb Gas approach

TL;DR

This paper analytically derives the large fluctuation rate functions for extreme eigenvalues of Jacobi ensembles using Coulomb gas methods, validated by known results and simulations.

Contribution

It introduces an analytical approach to compute the joint distribution and rate functions of extreme eigenvalues in Jacobi ensembles via Coulomb gas techniques.

Findings

Derived explicit rate functions for large fluctuations of eigenvalues.

Validated analytical results with known exact solutions.

Achieved good agreement between theory and numerical simulations.

Abstract

Using the Coulomb gas method and standard methods of statistical physics, we compute analytically the joint cumulative probability distribution of the extreme eigenvalues of the Jacobi-MANOVA ensemble of random matrices, in the limit of large matrices. This allows us to derive the rate functions for the large fluctuations to the left and the right of the expected values of the smallest and largest eigenvalues analytically. Our findings are compared with some available known exact results as well as with numerical simulations finding good agreement.