Moment determinants as isomonodromic tau functions

TL;DR

This paper demonstrates that a broad class of determinants related to moments of semiclassical functionals serve as isomonodromic tau functions, linking their vanishing to the solvability of associated Riemann-Hilbert problems and connecting to random matrix models.

Contribution

It establishes that these determinants are tau functions for isomonodromic families, unifying various classes of determinants under a common integrable systems framework.

Findings

Determinants are tau functions for isomonodromic deformations.

Vanishing tau-function indicates obstructions to Riemann-Hilbert problem solvability.

Includes determinants relevant to orthogonal polynomials and random matrix models.

Abstract

We consider a wide class of determinants whose entries are moments of the so-called semiclassical functionals and we show that they are tau functions for an appropriate isomonodromic family which depends on the parameters of the symbols for the functionals. This shows that the vanishing of the tau-function for those systems is the obstruction to the solvability of a Riemann-Hilbert problem associated to certain classes of (multiple) orthogonal polynomials. The determinants include Haenkel, Toeplitz and shifted-Toeplitz determinants as well as determinants of bimoment functionals and the determinants arising in the study of multiple orthogonality. Some of these determinants appear also as partition functions of random matrix models, including an instance of a two-matrix model.