Random matrices and integrable systems

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This program concerns the interface between the theory of integrable systems and the spectral theory of random matrices. The motivation for this work derives from the very wide applicability of results from random matrix theory in diverse areas of physics. Amongst the most prominent are: its relation to growth phenomena, both deterministic and stochastic; the computation of correlation functions in super Yang-Mills theory; problems of graphical enumeration, two dimensional quantum gravity, and conformal models.     The tools used are drawn mainly from the inverse spectral methods developed in the study of integrable systems. The relation between the two topics lies in the fact that parameters appearing in the measure defining random matrix ensembles and those determining the support set may be viewed as dynamical deformation parameters. The statistical quantities studied in random matrix theory satisfy systems of partial differential and difference equation with respect to these parameters that turn out to be exactly of the type entering in the theory of integrable systems.    One main objective of this program is to extend to the multi-matrix setting the nonlinear asymptotic methods that have been so successfully applied in studying the large N asymptotics of the one-matrix case. We are in a position to extend this approach to the multi-matrix case since the Riemann-Hilbert characterization of the underlying biorthogonal polynomials is known, and so is the "model solution", in terms of the usual analytic tools used in the function and deformation theory on Riemann surfaces.   Another approach that is being developed is based on free fermion quantum fields, stemming from the early work of the Kyoto school on integrable hierarchies. This method has been successfully used so far for computing character expansions of two-matrix and complex matrix integrals, as well as rational correlations functions of characteristic polynomials. These results should admit extensions to more general matrix groups, as well as to multi-matrix models.