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Critical phenomena in random matrix theory and integrable systems

Final Report Summary - CRAMIS (Critical phenomena in random matrix theory and integrable systems)

The objective of the project was to describe critical phenomena which occur in random matrix theory, integrable partial differential equations and Toeplitz determinants. Solutions of Painlevé equations play an important and universal role in the description of these critical phenomena. A first highlight of the project is a detailed description of the asymptotic behaviour of Toeplitz determinants with merging Fisher-Hartwig singularities in terms of special solutions of the fifth Painlevé equation, by the PI in collaboration with Igor Krasovsky. These results have led to rigorous proofs of a long-standing conjecture of Dyson and part of a recent conjecture of Fyodorov and Keating, and have also found applications in Gaussian processes. A second highlight consists of a series of results related to eigenvalue statistics in random matrix ensembles. We mention in this context the description of macroscopic eigenvalue behaviour in Muttalib-Borodin ensembles, the study of thinned random matrix ensembles, and large gap asymptotics for product random matrices.
From a methodological point of view, the main contributions of the project lie in the field of asymptotic analysis, in particular in the development of novel Riemann-Hilbert and steepest descent techniques.