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Random matrix theory and physical applications

The theory of random matrices studies the properties of matrices whose entries are random variables. The degree of independence and how the entries are distributed determine the type of random matrix ensemble to which the matrices belongs. Random matrices made their first appearance in physics in the 1950's (and earlier on, in the study of multivariate statistics), to explain the statistical properties of scattering resonances observed in nuclear reactions. Random-matrix theory (RMT) has since been found to be connected with a number of different mathematical areas and applications in many branches of physics have emerged and been consolidated as well.

We study mathematical properties of random matrices and its relationships with both analytical and algebraic problems involving structured matrices (such as Toeplitz and Hankel matrices) and problems in algebraic combinatorics, involving symmetric polynomials, such as Schur polynomials.

The other main line involves physical applications, mostly, but not exclusively, in the study of quantum field theories with gauge symmetry. In this area, random matrix theory tools are useful to compute observables of the theories, such as partition functions and Wilson loops, and also to analyze phase transitions in the theories. The range of theories where this tools can be applied is broad, as it includes supersymmetric theories in a number of dimensions, topological theories such as Chern-Simons theory and also bosonic theories in low dimensions.