Steklov Institute of Mathematics at St.Petersburg

PREPRINT 04/2005

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This preprint was accepted February 12, 2005

The main goal of this paper is to construct and develop a new
general method of computing gap probabilities in models of random
matrix type in terms of solutions of Painleve equations and of
more general isomonodromy deformations of differential and
difference linear systems with rational coefficients.

The method is based on the theory of integrable operators and
associated Riemann-Hilbert problems. In the first chapter
previously known results are applied to solving a problem of
harmonic analysis on the infinite-dimensional unitary group in
terms of the tau-function of Schlesinger equations which in simple
cases reduce to the classical Painleve equations.
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