Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem

Abstract

We consider biorthogonal polynomials that arise in the study of a generalization of two--matrix Hermitian models with two polynomial potentials V_1(x), V_2(y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials (`windows'), of lengths equal to the degrees of the potentials, satisfy systems of ODE's with polynomial coefficients as well as PDE's (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced.Comment: v1:41 pages, 5 figures, 1 table. v2:Typos and other errors corrected. v3: Some conceptual changes, added appendix and two figures v4: Minor typographical changes, improved figures. v5: updated version (submitted) 49 pages, 7 figures, 1 tabl