The probability distribution of spectral moments for the Gaussian beta-ensembles

TL;DR

This paper derives the joint probability distribution of the first two spectral moments for Gaussian beta-ensembles in any dimension, using a novel approach based on invariants of the domain.

Contribution

It introduces a new method to derive spectral moment distributions for Gaussian beta-ensembles, differing from previous techniques and providing potential for further insights.

Findings

Derived joint distribution for first two spectral moments in any dimension

Utilized invariants of the domain in spectral moment analysis

Discussed challenges in extending to higher spectral moments

Abstract

We derive the joint probability distribution of the first two spectral moments for the G$\beta$E random matrix ensembles in N dimensions for any N. This is achieved by making use of two complementary invariants of the domain in $\mathbb{R}^N$ where the spectral moments are defined. Our approach is significantly different from those employed previously to answer related questions and potentially offers new insights. We also discuss the problems faced when attempting to include higher spectral moments.