### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 232-268 |

Number of pages | 37 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 152-153 |

DOIs | |

State | Published - May 15 2001 |

### Keywords

- Biorthogonal polynomials
- Riemann-Hilbert problem
- Two-matrix model

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics