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

2006/03/31 by Saugata Ghosh
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebra over a field #Christoffel symbols #Classical orthogonal polynomials #Gaussian #Mathematics #Matrix (chemical analysis) #Orthogonal polynomials #Physics #Pure mathematics #Random Matrices and Applications #Random matrix #Skew #Skew-symmetric matrix #Square matrix #Symmetric matrix #Symplectic geometry #math-ph #math.MP

paper · pdf · doi:10.1088/0305-4470/39/28/s02

published as J. Phys. A: Math. Gen. 39 (2006) 8775-8782 · 8 pages

openalex publication_date 2006/06/27 · arxiv created 2006/06/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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

Citations

• A concise expression for the ODE's of orthogonal polynomials
• Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices
• Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices