Two matrix models and biorthogonal polynomials

Bertrand Eynard

(Joint series with Marco Bertola and John Harnad)


Lecture 1. (Monday June 27, B. Eynard) Introduction to two-matrix and multi-matrix models.
We will review definitions and motivations for the 2-matrix model, and explain the relationship with biorthogonal polynomials, showing how to compute eigenvalue correlations in terms of kernels. We will also comment on generalizations of this model: the chain of matrices. The main goal is to be able to compute large N correlation functions, i.e. we need the kernels in the large N limit, and in various scaling regimes. Thus, one needs large N asymptotics for large N biorthogonal polynomials, and we will introduce the recursion and differential equations that are needed for that purpose.

Lecture 2. (Tuesday June 28, M. Bertola) Differential equations, finite N spectral curve and duality.
The recursion relations and the differential and deformation equations satisfied by paired sequences of polynomials that are biorthogonal with respect to exponential polynomial measures will be derived, as well as the generalized Christoffel-Darboux relations. The differential-deformation-recursion relations will be expressed on finite "windows" of biorthogonal polynomials, and their integral transforms, using the method of "folding". It will be shown that these differential systems are in duality, in the sense that they all share the same spectral curve, and that they represent dual pairs of differential operators with polynomial coefficients, whose generalized monodromy (i.e. Stokes data at infinity) is preserved under both the recursions and the deformations induced by changes of the polynomial potentials defining the biorthogonality measures.

Lecture 3. (Wednesday June 29, J. Harnad) Integral representations of the fundamental systems.
For paired sequences of polynomials that are biorthogonal with respect to measures determined by polynomial potentials, integral representations of the fundamental systems satisfying the recursion and differential equations will be derived. These will be shown to be bilinearly paired with integral representations of the fundamental systems associated to the dual sequence of Fourier-Laplace transforms of the polynomials. They satisify differential systems which determine the spectral curve and deformation equations that determine the partition function. The integral representations lead to a Riemann-Hilbert characterization determining the biorthogonal sequences, which may be used as the starting point in large N asymptotic analysis. These integral representations will be related to another one, recently proposed by Kujlaars and Mclaughlin, based on multi-orthogonal polynomials, in which the jump matrices are nonconstant functions.

Lecture 4. (Wednesday June 29, M. Bertola) Dual Riemann-Hilbert approach to biorthogonal polynomials.
The integral representations of the fundamental systems of solutions to the differential-deformation-recursion relations introduced in the previous lecture will be analyzed via the steepest descent method to deduce the large argument asymptotics of the biorthogonal polynomial sequences and their integral transforms. Together with the local jump discontinuitites across the integration contours, this provides a Riemann-Hilbert characterization of these fundamental systems and their duals, giving a starting point for the Riemann-Hilbert approach to the study of large N asymptotics of biorthogonal polynomials.

Lecture 5. (Thursday June 30, B. Eynard) Formal large N expansion for the 2-matrix model and algebraic geometry.
The formal 2-matrix model can be defined as a combinatoricial generating function for enumerating discrete surfaces. By definition, this model has a well defined 1/N expansion. We will show in this lecture how to solve recursively the Schwinger-Dyson equation, and find a formula for all terms in the 1/N expansion. To leading order, the Schwinger-Dyson equation is an algebraic equation, and all the other orders can be written in terms of geometric properties of the leading order algebraic curve. To do this, we will introduce some basics of analysis on algebraic curves.

Lecture 6. (Friday July 1, B. Eynard) Large N asymptotics of biorthogonal polynomials.
We will present a conjecture for the asymptotics of biorthogonal polynomials. This conjecture is in terms of algebraic geometry of the large N limit of the spectral curve.