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An Isaac Newton Institute Workshop

Painleve Equations and Monodromy Problems: Recent Developments

Matrix integrals as isomonodromic tau functions

Author: John Harnad (Centre de recherches mathematiques, and Concordia University)

Abstract

It is well known that the tau functions associated to special solutions of the Painleve equations may be expressed as matrix integrals (e.g. gap probabilities for sine kernel, airy kernel or Bessel kernel determinantal ensembles). The partition functions for many types of matrix models are also known to be isomondromic tau functions, as are various types of correlation functions. More generally, for a wide variety of generalized orthogonal polynomial (Christoffel-Darboux kernel) ensembles, with orthogonality support taken on quite general curve segments in the complex plane, the matrix integrals representing partition functions, gap probabilities and expectation values of spectral invariant functions can all be interpreted on the same footing, and shown to be isomonodromic tau functions. This result also extends to two-matrix integrals, which are associated with the isomonodromic systems corresponding to sequences of biorthogonal polynomials.

(This talk is based on joint work wih: Marco Bertola, Bertrand Eynard and Alexander Orlov)

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