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

We give a rigorous construction of complete families of biorthonormal polyn omials associated to a planar measure of the form e(-n(V(x)+W(y)-2xxy))dx d y for polynomial V and W. We are further able to show that the zeroes of th ese polynomials are all real and distinct. A complex analytical constructio n of the biorthonormal polynomials is given in terms of a non-local Riemann -Hilbert problem which, given our prior result, provides an avenue for deve loping uniform asymptotics for the statistical distributions of these zeroe s 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 indu ced by linear deformations of V and W coincide with a semi-infinite general ization of the completely integrable full Kostant-Toda lattice. This connec tion could be relevant for understanding aspects of scaling limits for the multi-matrix model. (C) 2001 Elsevier Science B,V, All rights reserved.