The PDEs of biorthogonal polynomials arising in the two-matrix model

Abstract

The two-matrix model can be solved by introducing bi-orthogonal polynomials. In the case the potentials in the measure are polynomials, finite sequences of bi-orthogonal polynomials (called "windows") satisfy polynomial ODEs as well as deformation equations (PDEs) and finite difference equations (Delta-E) which are all Frobenius compatible and define discrete and continuous isomonodromic deformations for the irregular ODE, as shown in previous works of ours. In the one matrix model an explicit and concise expression for the coefficients of these systems is known and it allows to relate the partition function with the isomonodromic tau-function of the overdetermined system. Here, we provide the generalization of those expressions to the case of bi-orthogonal polynomials, which enables us to compute the determinant of the fundamental solution of the overdetermined system of ODE+PDEs+Delta-E.

20 pages v1 18 Nov 2003; v2 9 Jan 2004: trivial Latex mistake corrected

References in corpus (4)

• Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem
• Partition functions for Matrix Models and Isomonodromic Tau functions
• Bilinear semi-classical moment functionals and their integral representation
• The Riemann-Hilbert Problem for the Bi-Orthogonal Polynomials

Cited by in corpus (12)

• Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure
• The partition function of the two-matrix model as an isomonodromic tau-function
• Universality in the two matrix model with a monomial quartic and a general even polynomial potential
• Mixed correlation function and spectral curve for the 2-matrix model
• A critical phenomenon in the two-matrix model in the quartic/quadratic case
• Peakons and Cauchy Biorthogonal Polynomials
• Skew-orthogonal polynomials, differential systems and random matrix theory
• Transitions between critical kernels: from the tacnode kernel and critical kernel in the two-matrix model to the Pearcey kernel
• Generalized Christoffel-Darboux formula for classical skew-orthogonal polynomials
• The Hermitian two matrix model with an even quartic potential
• Painlevé kernels in Hermitian matrix models
• Universality in the two matrix model: a Riemann-Hilbert steepest descent analysis