Linear statistics of random matrices and log-gases
Time: Fri 2023-06-02 13.00
Location: D2, Lindstedtsvägen 5, Stockholm
Subject area: Mathematics
Doctoral student: Klara Courteaut , Matematik (Avd.)
Opponent: Christian Webb, University of Helsinki
Supervisor: Kurt Johansson, Matematik (Avd.)
This thesis is concerned with point processes arising in Random Matrix Theory. It is a compilation thesis: it consists of an introduction and three research papers.
In Paper A and Paper B, we study random matrices from the classical compact groups, namely orthogonal, unitary, and symplectic matrices distributed according to Haar measure. We consider the moments of the empirical spectral measure, i.e. the trace of the powers of the matrices. These are known to converge in distribution, as the dimension of the matrices tends to infinity, to independent, normal random variables. We show that the convergence is especially fast as the total variation distances to the limiting Gaussians decay faster than exponentially.
Paper A is devoted to the multivariate case for orthogonal and symplectic matrices: we study the total variation between a vector filled with the trace of the powers of the matrix, and one filled with independent normal random variables. We obtain an explicit upper bound on the total variation which depends on the highest power and the dimension of the matrix, and which, as a function of their ratio, decays faster than exponentially.
In Paper B we consider a single trace at a time. This allows to study the trace of a higher power, up to the dimension of the matrix in the unitary case. We show a transition from a super-exponential decay, for small powers, to a decay of Berry-Esseen type, for high powers. The argument also gives more precise bounds. We obtain a first lower bound when the power equals one and determine the exact asymptotics of the L2 distance.
In Paper C we consider a Coulomb gas on a sufficiently regular Jordan curve in the plane, at a general temperature. This is a generalization of the Circular β-Ensemble. We derive the asymptotics of the partition function from those of the Laplace transform of linear statistics. The asymptotic formula is expressed, among other things, in terms of the Grunsky operator associated with the exterior conformal mapping for the curve.