TY - JOUR

T1 - Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model

AU - Ercolani, Nicholas M.

AU - McLaughlin, Kenneth T.R.

N1 - Funding Information:
The authors would like to gratefully acknowledge the support they received from NSF Grants DMS-9626306 and DMS-9970328.

PY - 2001/5/15

Y1 - 2001/5/15

N2 - 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.

AB - 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.

KW - Biorthogonal polynomials

KW - Riemann-Hilbert problem

KW - Two-matrix model

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U2 - 10.1016/S0167-2789(01)00173-7

DO - 10.1016/S0167-2789(01)00173-7

M3 - Article

AN - SCOPUS:0035873520

VL - 152-153

SP - 232

EP - 268

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

ER -