Generalized Christoffel-Darboux formula for skew-orthogonal polynomials and random matrix theory

TL;DR

This paper derives a generalized Christoffel-Darboux formula for skew-orthogonal polynomials and applies it to analyze level density and correlations in Gaussian orthogonal and symplectic ensembles of random matrices.

Contribution

It introduces a new generalized Christoffel-Darboux formula for skew-orthogonal polynomials and uses it to derive key spectral properties of GOE and GSE.

Findings

Derived the generalized Christoffel-Darboux formula for SOP

Provided an alternative derivation of level density for GOE and GSE

Enhanced understanding of spectral correlations in random matrix ensembles

Abstract

We obtain generalized Christoffel-Darboux (GCD) formula for skew-orthogonal polynomials (SOP). Using this, we present an alternative derivation of the level density and two-point function for Gaussian orthogonal ensembles (GOE) and Gaussian symplectic ensembles (GSE) of random matrices.