Biorthogonal Laurent polynomials, Toeplitz determinants, minimal Toda orbits and isomonodromic tau functions

Abstract

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish corresponding Christoffel-Darboux formulae . For all these classes of polynomials a 2x2 system of Differential-Difference-Deformation equations is analyzed in the most general setting of pseudo measures with arbitrary rational logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann sphere. The corresponding isomonodromic tau function is explicitly related to the shifted Toeplitz determinants of the moments of the pseudo-measure. In particular the results imply that any (shifted) Toeplitz (Hankel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic tau function.

35 pages, 1 figure

References in corpus (7)

• 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
• Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems
• Partition functions for Matrix Models and Isomonodromic Tau functions
• Bilinear semi-classical moment functionals and their integral representation
• Inverse moment problem for elementary co-adjoint orbits

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• Boutroux curves with external field: equilibrium measures without a minimization problem
• The partition function of the two-matrix model as an isomonodromic tau-function
• Algebro-Geometric Finite-Gap Solutions of the Ablowitz-Ladik Hierarchy
• Moment determinants as isomonodromic tau functions
• Peakons and Cauchy Biorthogonal Polynomials
• Biorthogonal polynomials for 2-matrix models with semiclassical potentials
• Effective inverse spectral problem for rational Lax matrices and applications
• Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights
• Matrix biorthogonal polynomials in the unit circle: Riemann-Hilbert problem and matrix discrete Painleve II system
• Bi-orthogonal systems on the unit circle, regular semi-classical weights and the discrete Garnier equations
• The Ablowitz-Ladik Hierarchy Revisited
• Bi-orthogonal systems on the unit circle, Regular Semi-Classical Weights and Integrable Systems - II