Publication : t02/097

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

Bertola M. (Centre de Recherches Mathématiques (CMR), Université de Montréal C.P. 6128, succ. centre ville, Montréal, Québec H3C 3J7, CANADA)
Eynard B. (CEA, DSM, SPhT (Service de Physique Théorique), F-91191 Gif-sur-Yvette, FRANCE)
Harnad J. (CRM et Université Concordia, Montréal, CANADA)
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 $V_1$ and $V_2$, 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.
Année de publication : 2003
Revue : Commun. Math. Phys. 243 193-240 (2003)
DOI : 10.1007/s00220-003-0934-1
Preprint : arXiv:nlin.SI/0208002
Numéro Exterieur : CRM-2852_(juillet_2002)
Langue : Anglais

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