Congratulations to Ferenc Balogh of the John Abbott College Mathematics department. He was awarded a research grant from the Fonds de recherche du Québec – Nature et technologies (FRQNT).
Ferenc’s research project: “Asymptotiques des statistiques des valeurs propres dans les matrices aléatoires planaires“, proposes to include two JAC students to help in this research project.
To read the summary of this research proposal, click on the link below.
“A random matrix is an array of random numbers drawn from a prescribed probability distribution. The classical models of Hermitian random matrices are defined by densities on the space of Hermitian matrices that are invariant under the action of the unitary group.
The statistics of Hermitian matrices are well understood: the correlation functions of their eigenvalues are expressible in terms of orthogonal polynomials, and their asymptotic expansion in the large matrix size limit is governed by the equilibrium measure that minimizes the energy in an electrostatic variational problem. The supports of the orthogonality measure and the equilibrium measure are both real when the model is Hermitian. The eigenvalues behave like electrically charged particles on the real line, and by this analogy one can use them to model various systems with a built-in repulsion between its components, like the distribution of cars parked on a street, perched birds on a wire, the departure times of autobuses in a self-managing transport network, or the boarding times of passengers of an aircraft.
Thanks to the Riemann-Hilbert method, developed by Deift and Zhou, it is possible to find the asymptotic expansions of the orthogonal polynomials which may be used to obtain universal limits of the correlation functions in different scaling limits, which only depend on the underlying symmetries of the model. My research work focuses on the relation between the asymptotics of orthogonal polynomials and the equilibrium measures for planar random matrix models for which the eigenvalues are not confined to the real axis. These models are motivated by their potential applications to phenomena on land or sea where a repulsive force is present between the components of a system.
The Riemann-Hilbert approach for Hermitian matrices is not directly applicable to planar matrix models, a general method to get the asymptotic expansions of the correlation functions is yet to be found. To date, this problem is solved only in a handful of very special examples.
With my previous work on the subject as starting point, I propose the present project using various techniques from potential theory, conformal mappings, orthogonal polynomials, quadrature domains, and Fredholm determinants to find new universal statistical laws for planar random matrices.”