Fluctuations of random matrix eigenvalues and disordered systems

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This NSERC Discovery Grant is focused on studying fluctuations of eigenvalues of random matrices and random growth models. The classical central limit theorem states that the distribution of large sums of independent random variables looks approximately Gaussian, regardless of the distribution of the individual random variables. That is, as the system size (the number of random variables) tends to infinity, the sum converges to a universal probability distribution that does not depend on the microscopic details of the system. Today, probability theory is now discovering and trying to understand a new form of universality, that characterizes large correlated systems instead of independent ones. One of the first forms of universality was put forward by the physicist Wigner, who postulated that the eigenvalues of random matrices were a good model for the energy levels of large quantum systems, such as heavy atoms. Since Wigner's original work, random matrix statistics have been discovered or are conjectured to appear in a wide range of seemingly unrelated settings, including in the zeros of the Riemann zeta function, in the quantization of chaotic classical systems, and in random growth models in the Kardar-Parisi-Zhang universality class. The proposed research has 3 specific Research Aims that will further our understanding of this phenomena of universality by: (1) Investigating the behaviour of the maximum of the characteristic polynomial of solvable random matrix models and further proving universality for general matrices, as well as other extremal statistics. (2) Studying eigenvalue dynamics in new settings, including dynamics on the complex plane as well as those of unitary matrices. (3) Studying general random growth models by importing the ideas and analytic methods of random matrix theory, rather than relying on exact formulas.  Random matrix eigenvalues have been proposed as a probabilistic model for the zeros of the Riemann zeta function, and the study of the maximum of the characteristic polynomial and other extremal eigenvalue statistics is motivated in part by investigating this correspondence (Research Aim 1). Stochastic eigenvalue dynamics have emerged as an important tool for understanding random matrix eigenvalues, and provide a link between random matrices and interacting particle systems (Research Aim 2). Large classes of random growth models (the KPZ universality class) are conjectured to enjoy a universal behaviour in the limit of large system size.  The majority of models for which the behaviour can be proven are special cases that can be treated by exact formulas, which falls far short of the broadest visions of universality (Research Aim 3). The expected outcome of the proposed research will be to develop new techniques and methods that will increase our understanding of the universality of correlated systems, and further establish the links between random matrix theory and other areas of mathematics and science.