Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model

We give a rigorous construction of complete families of biorthonormal polynomials associated to a planar measure of the form e−n(V(x)+W(y)−2τxy)dx dy for polynomial V and W. We are further able to show that the zeroes of these polynomials are all real and distinct. A complex analytical construction of the biorthonormal polynomials is given in terms of a non-local Riemann–Hilbert problem which, given our prior result, provides an avenue for developing uniform asymptotics for the statistical distributions of these zeroes as n becomes large. The biorthonormal polynomials considered here play a fundamental role in the analysis of certain random multi-matrix models. We show that the evolutions of the recursion matrices for the polynomials induced by linear deformations of V and W coincide with a semi-infinite generalization of the completely integrable full Kostant–Toda lattice. This connection could be relevant for understanding aspects of scaling limits for the multi-matrix model.