The principal goal of the research project is to address some of the new analytical questions of the theory of integrable systems which emerged from the recent developments in random matrix theory and in the related areas of exactly solvable quantum models. The problems under consideration include the analytical description of the Painlev'e - type universalities in random matrix theory and the related study of special families of Painlev'e transcendents and generalized Painlev'e transcendents arising in the critical asymptotics in random matrices, statistical mechanics and in the enumerative topology. The third direction of the project is concerned with the analytical investigation of the crossover phenomena in the classical theory of Toeplitz and Hankel determinants, i.e. with the study of the Painlev'e-type transition behavior between different non-critical asymptotic regimes exhibited by large size Toeplitz and Hankel determinants. Each of the above mentioned directions is represented by a collection of concrete problems, and they are proposed to be investigated within the same analytical framework, which is the Riemann-Hilbert method.
The theory of integrable systems is an expanding area which plays an increasingly important role as one of the principal sources of new analytical and algebraic ideas for many branches of modern mathematics and theoretical physics. Simultaneously, it provides an efficient analytical tool for study of some of the fundamental mathematical models arising in modern nonlinear science and technology. The new areas where the analytic techniques of integrable systems become more and more common include random matrices, quantum field and string theories, enumerative topology, stochastic processes, number theory, and, most recently, entanglement in quantum chain systems which is expected to play prominent role in future quantum computing technology. The problems considered in the proposal have direct connections with the mentioned disciplines. In particular, part of the proposal related to the Toeplitz and Hankel determinants is primary motivated by the needs of the analytical theory of quantum entanglement and by the challenges of analytical description of critical behavior in important stochastic and quantum field models. This part of the proposal is also essential for testing the random matrix predictions in number theory. Proposed study of the Painlev'e - type universalities and the families of special Painlev'e functions has direct relation to enumerative topology and string theory. Success in achievement of the proposal's goals will have a notable impact on research in all these areas.
The term ``Integrable Systems'' usually refers to mathematical objects, most often differential equations, with special symmetry properties which allow to study them in a very detailed way and sometimes even to solve them in a closed form. The class of integrable systems includes several fundamental equations of nature, and the mathematical foundations of integrable systems go back to classical works of Liouville, Gauss, and Poincare. In our days, the theory of integrable systems has become an expanding area which plays an increasingly important role as one of the principal sources of new analytical and algebraic ideas for many branches of modern mathematics and theoretical physics. Simultaneously, it provides an efficient analytical tool for the study of some of the fundamental mathematical models arising in modern nonlinear science and technology. The proposal was devoted to the investigation of the analytical aspects of the theory of integrable systems related to the random matrix theory, orthogonal polynomials, and to the theory of Toeplitz matrices and determinants. The main results of this three year research program are the following. (i) A new universality class for unitary matrix ensembles whose weight has a root singularity at the edge of the support of equilibrium measure has been described. (ii) Higher order analogs of the well known Tracy-Widom distribution corresponding to the higher universality classes in the edge behavior of the usual unitary ensembles, have been obtained. (iii) The proof of the long-standing conjecture of Basor and Tracy concerning the asymptotics of Toeplitz determinants with a Fisher-Hartwig symbol of general form, has been obtained (iv) The asymptotic behavior of Toeplitz determinants, as the symbol is deformed from smooth SzegH{o} to singular Fisher-Hartwig type, has been obtained. The result generalizes the celebrated scaling theory for 2D Ising model of Wu, McCoy, Tracy and Barouch. (v) A detailed description of the spectrum of large Toeplitz matrices has been obtained Previously known results (Bottcher et.al.) have been generalized and some of important conjectures concerning the large Toeplitz matrices (Sobolev et. al.) have been proven. (vi) The asymptotic analysis of the block Toeplitz determinants associated with the von Neumann entropy of a large class of quantum spin chains has been performed. The research which led to the above listed results was primarily motivated by the intrinsic needs of the modern theory of integrable systems, notably by the needs of development of the novel analytic apparatus, the so-called Riemann-Hilbert Method. In what follows, the broader impact of the work done in the project is described. Results concerning the new universality classes in random matrix theory: These results contribute to the development of this important new area whose probabilistic laws govern the statistical properties of large systems which do not obey the standard laws of classical probability. Such systems appear in many different scientific areas, and in applied science and technology. In particular, such systems include heavy nuclei, polymer growth, high-dimensional data analysis, and certain percolation processes. Results concerning the asymptotic analysis of Toeplitz determinants: These results contribute to the study of critical phenomena in statistical mechanics, i.e., to the study of phase transitions - an enduring, fundamental problem in the field. Specifically, the proof of the Basor-Tracy conjecture has already made it possible to justify various asymptotic formulae that appeared previously in the physical literature on the basis of this conjecture. These results have also been used in condensed matter physics in recent studies of quantum many-body systems away from equilibrium and in testing random matrix predictions in number theory. Results concerning block Toeplitz determinants: The asymptotic analysis of block Toeplitz determinants is directly linked to the analytical theory of quantum entanglement in the spin chains. The quantum spin chains are expected to play the key role in a practical realization of quantum computing.
