Schur expansion of random-matrix reproducing kernels

Abstract

We give expansions of reproducing kernels of the Christoffel-Darboux type in terms of Schur polynomials. For this, we use evaluations of averages of characteristic polynomials and Schur polynomials in random matrix ensembles. We explicitly compute new Schur averages, such as the Schur average in a $q$-Laguerre ensemble, and the ensuing expansions of random matrix kernels. In addition to classical and $q$-deformed cases on the real line, we use extensions of Dotsenko-Fateev integrals to obtain expressions for kernels on the complex plane. Moreover, a known interplay between Wronskians of Laguerre polynomials, Painlevé tau functions and conformal block expansions is discussed in relationship to the Schur expansion obtained.

26 pages. v2: minor revision, references added. Published version

References in corpus (7)

• Kernel methods in machine learning
• On the complete perturbative solution of one-matrix models
• Chern-Simons matrix models and Stieltjes-Wigert polynomials
• Characteristic polynomials in real Ginibre ensembles
• Coulomb integrals for the SL(2,R) WZNW model
• Elliptic $q,t$ matrix models
• Multivariable Christoffel-Darboux Kernels and Characteristic Polynomials of Random Hermitian Matrices