This preprint was accepted February 12, 2005
ABSTRACT: 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.[Full text: (.ps.gz)]