The Riemann-Hilbert Problem for the Bi-Orthogonal Polynomials

Abstract

For the bi-orthogonal polynomials with the third degree polynomial potential functions, the 3 x 3 matrix Riemann-Hilbert problem is explicitly constructed. The developed approach admits an extension to the bi-orthogonal polynomials with arbitrary polynomial potentials.

LaTeX, 15 pages. In compare to the original version, the derivation of the matrix wave function is simplified; some comments and references are added

Cited by in corpus (12)

• Normal random matrix ensemble as a growth problem
• Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy
• The PDEs of biorthogonal polynomials arising in the two-matrix model
• 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
• Monodromy approach to the scaling limits in the isomonodromy systems
• Skew-orthogonal polynomials, differential systems and random matrix theory
• Average characteristic polynomials in the two-matrix model
• Generic critical points of normal matrix ensembles
• Universality and critical behavior in the chiral two-matrix model