Painleve Transcendents and their appearance in Physics and Random Matrix Theory

The classical Painleve' equations have been playing an increasingly important  role in physics since 1970s-1980s works of Barouch, McCoy, Tracy, and Wu, and of Jimbo, Miwa, Mori and Sato devoted to the quantum correlation functions. Since the early nineties, the Painleve' transcendents have become a major player in the theory and applications of Random Matrices as well (the pioneering works of Bre'zin and Kazakov, Duglas and Shenker, Gross and Migdal, Mehta and Mahoux, and Tracy and Widom). In this talk we will try to review these, and also some of the more recent results concerning Painleve' transcendents and their appearance in random matrices, statistical mechanics and quantum field theory. We will present a unified point view on the subject based on the Riemann-Hilbert method. The emphasis will be made on the various double scaling limits related to the universal properties of random matrices for which Painleve' functions provide an adequate ''special function environment''.

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