Generalized Christoffel-Darboux formula for classical skew-orthogonal polynomials

TL;DR

This paper extends the Christoffel-Darboux formula to skew-orthogonal polynomials associated with classical weights, enabling analysis of eigenvalue correlations and universality in random matrix ensembles.

Contribution

It introduces a generalized Christoffel-Darboux formula for skew-orthogonal polynomials in quaternion space, applicable to classical weights, and explores universality in eigenvalue correlations.

Findings

Derived GCD formul extaeor kernel functions in random matrix ensembles.

Proved universality of eigenvalue correlations in the bulk spectrum.

Established a mapping between skew-orthogonal functions in orthogonal and symplectic ensembles.

Abstract

We show that skew-orthogonal functions, defined with respect to Jacobi weight $w_{a,b}(x)={(1-x)}^a{(1+x)}^b$, $a$, $b>-1$, including the limiting cases of Laguerre ($w_{a}(x)=x^{a}e^{-x}$, $a > -1$) and Gaussian weight ($w(x)=e^{-x^2}$), satisfy three-term recursion relation in the quaternion space. From this, we derive generalized Christoffel-Darboux (GCD) formul\ae\ for kernel functions arising in the study of the corresponding orthogonal and symplectic ensembles of random $2N\times 2N$ matrices. Using the GCD formul\ae we calculate the level-densities and prove that in the bulk of the spectrum, under appropriate scaling, the eigenvalue correlations are universal. We also provide evidence to show that there exists a mapping between skew-orthogonal functions arising in the study of orthogonal and symplectic ensembles of random matrices.