Random matrices and related models

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In a one-lane traffic situation, one slow car may delay a whole platoon of cars behind. This situation is an example of randomly interacting systems in which particles (cars) have random amounts of time to move, and one particle's movement influences other particles in some distance. The spread of fire, the cavity of melting ice, and growth of bacterial colonies also have similar features. In this project, the principal investigator studies some of the underlying fundamental mathematical properties for a class of simplified models representing such random interactions. In particular, the principal investigator studies how the randomness of the particles movement impact the global characteristics of the system.The principal investigator will study the fluctuations of several random systems. They include spin glass, random matrices, and interacting particle systems. The overarching goal is to explore the scope and limitation of the universal distribution functions that arise from random matrix theory and related models and also uncover new universal distributions. Specifically, the principal investigator studies 2-spin spherical Sherrington-Kirkpatrick model of spin glass, totally asymmetric simple exclusion process on a ring, and the Airy process. the principal investigator uses the exactly solvable nature of these models extensively to obtain precise asymptotic results. Riemann-Hilbert problems and coordinate Bethe ansatz method are some of the tools used.