Random matrix models and their applications to statistical mechanics

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This research project is directed on basic problems of the theory of random matrix models and their applications to statistical mechanics. The project concerns with different conjectures of universality of the scaling limits of eigenvalue correlation functions and with exactly solvable models of statistical physics. This includes: (a) the development of the Riemann-Hilbert (RH) approach to random matrix models with external source; (b) the semiclassical asymptotics and the RH approach to multi-matrix models; (c) the exact solution of the six-vertex model with domain wall boundary; (d) the study of the phase separation in the six-vertex model, and others.The project has an interdisciplinary character and it lies on the frontier between physics and mathematics. The problems of scaling and universality are central in many areas of modern science: theory of critical phenomena and phase transitions, statistical physics and quantum field theory, theory of quantum chaos, nonlinear dynamics, etc. This project is directed on the development of powerful mathematical methods to the problems of scaling and universality in the theory of random matrices and their applications to statistical physics. It involves methods of different areas of mathematics: analysis, theory of integrable systems, probability theory, semiclassical asymptotics, complex analysis, etc. The research project under consideration has direct applications to various physical problems: combinatorial asymptotics related to quantum gravity, exactly solvable models of statistical physics, spin systems on random surfaces, theory of critical phenomena and phase transitions, quantum chaos. Possible further applications include the theory of knots and links and related problems in molecular biology, growth models, statistical data analysis, and others.