Schur expansion of random-matrix reproducing kernels

TL;DR

This paper develops Schur polynomial expansions for random matrix kernels, utilizing characteristic polynomial averages and extending to complex plane cases, revealing connections with Painlevé functions and conformal blocks.

Contribution

It introduces explicit Schur average computations in new random matrix ensembles and derives kernel expansions on the complex plane, expanding the analytical toolkit for random matrix theory.

Findings

Derived Schur expansions for Christoffel-Darboux kernels.

Computed new Schur averages in q-Laguerre ensembles.

Connected kernel expansions with Painlevé tau functions.

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, Painlev\'e tau functions and conformal block expansions is discussed in relationship to the Schur expansion obtained.