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Universality is a fundamental aspect of our understanding of Nature. It means that many physical systems manifest the same behaviour independently of what the details of the interaction among their constituent elements are. For example, all macroscopic objects obey the laws of thermodynamics, while at the same time matter is built out of atoms. The conciliation of the macroscopic laws of thermodynamics with atomic physics has been a long-standing, fundamental and difficult challenge for scientists. In other words, on a macroscopic scale physical systems exhibit universality.Loosely speaking, Random Matrix Theory (RMT) can be thought of as a combination of linear algebra and the theory of probability. Each time a physical or mathematical process has a stochastic nature and its governed by linear equations, it is likely that it may be modelled by RMT. Indeed, random matrix models have fundamentally important applications in many branches of mathematics and physics such as combinatorics, complex systems, dynamical systems, growth problems, integrable systems, number theory, operator algebra, probability theory, quantum chaos, quantum field theory, quantum information, statistics, statistical mechanics, structural dynamics and telecommunications. The main feature that makes RMT a powerful tool in such wide range of applications is, once again, universality. In this context it means that for large matrix dimensions the local statistics of eigenvalues of random matrices depend only on the symmetries of the matrices, but are independent of the choice of the probability densities that govern their stochastic behaviour. Universality has been proved for a large class of Hermitian matrix models, but in its full generality it is still a conjecture.The main goal of this project is to prove universality in a vast class of non-Hermitian matrix models. Such ensembles of matrices find applications to growth problems, to the Hele-Shaw problem, especially in the vicinity of a critical point, and to semiclassical study of electronic droplets in the Quantum Hall regime. Non-Hermitian matrices do not have any symmetry constraints, with the exception that their elements must be real, complex or real quaternions respectively. The universality of the spectra of random matrices will be studied in all these three cases. Furthermore, universality will be investigated in the bulk as well in singular regions of the spectrum. It is expected that in these two cases the local spectral statistics will behave rather differently. However, they will still be universal, in the sense that they will depend only on the type of critical point but not on the probability distribution of the matrices.Finally, one of the main tools in the investigation of universality in non-Hermitian matrix models will be the asymptotic analysis of orthogonal polynomials in the complex plain using the dbar problem, which will have implications, for example, in the study of dispersionless multi-dimensional integrable systems and in the asymptotic analysis of integrable operators.
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