Moment determinants as isomonodromic tau functions

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.

24 pages

References in corpus (13)

• Matrices coupled in a chain. I. Eigenvalue correlations
• Duality, Biorthogonal Polynomials and Multi-Matrix Models
• Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem
• Random Words, Toeplitz Determinants and Integrable Systems. II
• Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions
• Partition functions for Matrix Models and Isomonodromic Tau functions
• Biorthogonal Laurent polynomials, Toeplitz determinants, minimal Toda orbits and isomonodromic tau functions
• Bilinear semi-classical moment functionals and their integral representation
• Zeros of the Jimbo, Miwa, Ueno tau function
• Random Matrix Theory and the Sixth Painlevé Equation
• Isomonodromic deformation of resonant rational connections
• Deformation analysis of matrix models
• Inverse moment problem for elementary co-adjoint orbits

Cited by in corpus (5)

• The dependence on the monodromy data of the isomonodromic tau function
• Biorthogonal ensembles with two-particle interactions
• Mixed moments of characteristic polynomials of random unitary matrices
• The partition function of the two-matrix model as an isomonodromic tau-function
• On a class of elliptic orthogonal polynomials and their integrability