Asymptotics of correlation functions for random matrices

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Random matrices is an area of analysis and probability which was discovered in pioneering works of Wigner and Dyson.Initially considered to describe the behaviour of many particle systems with unknown interactions via modelling these interactions by random variables, random matrices have since found a multitude of applicationsin physics, mathematics, and engineering. The main subject, still containing many unresolved problems, is the study of correlations between eigenvalues of random matrices when the size of the matrix is large. A typical such object is a probability that a given interval of the real line contains no eigenvalues of the random matrices. It is briefly called gap probability. This probability is expressed for many interesting ensembles ofrandom matrices as a Hankel or Toeplitz determinant. The question is to find explicit expansions of this object for the case of a large gap. This question is very easy to formulate, but obtaining the solution is a highly nontrivial task involvingdevelopment of appropriate methods. Very efficient ones rely on a so called Riemann-Hilbert approach, where the question is reduced to a problem for functions of complex variable.For some basic ensembles of random matrices, this question has been solved, but there are very important ones wherethe problem remains open. The purpose of the project is to address this question in one or two such open cases.Riemann-Hilbert steepest descent techniques used in this project have been initiated by Deift and Zhou 20 years ago, and have been developing ever since. They proved to be probably the most powerful tool of asymptotic analysis and helped to solve many important problems. Solution of a new problem using this approach also provides a development of the method itself, as new tricks have to be designed to overcome new obstacles.Thus, in addition to solving a new problem, we also contribute to the development of an important method.