Free convolutions and the Pearcey process in random matrix theory
Time: Fri 2022-10-28 10.30
Location: D2, Lindstedtsvägen 5, Stockholm
Subject area: Mathematics
Doctoral student: Philippe Moreillon , Matematik (Avd.)
Opponent: Professor Benoit Collins, Kyoto University, Japan
Supervisor: Kevin Schnelli,
The dissertation is in Random Matrix Theory, a field at the interface of probability theory, mathematical physics and operator algebras. First, we examine elementary questions that arise in Voiculescu’s Free Probability Theory of non-commutative random variables. We study qualitative properties of the free additive convolution of multi-cut measures. Second, we analyze a question in the field of determinantal point processes that can be formulated as a Riemann-Hilbert problem: The asymptotic analysis of the moment generating function of the universal Pearcey process. This compilation thesis consists of an introduction and three original research papers. In Paper A, we derive an upper and a lower bound on the number of connected components in the support of the free additive convolution of two multi-cut probability measures. In particular, we show that if they are supported on n and m intervals respectively, then their free additive convolution is supported on at most twice the product of n and m intervals. We further study the number of connected components in the support of the free additive convolution semi-group. The probability measures considered in this paper satisfy a power law behaviour with exponents strictly between −1 and 1. This class of measures includes most of the probability measures arising in random matrix theory. Besides methods of random matrix theory, we rely on special properties of analytic self-maps of the upper-half plane to reduce the question to a new combinatorics problem. In Paper B, we determine the asymptotics of the density of the free additive convolution semi-group at the endpoints and at any singularpoint in its support. In particular, we classify all the possible behaviors of the densities of these measures. We further study the free additive convolution of two multi-cut probability measures and show that its density decays either as a square root or as a cubic root at any endpoints of its support. In addition to the methods used in Paper A, we analyze perturbed solutions of non-linear equations of degree between two and three. In Paper C, we study the moment generating function of the Pearcey process on m intervals. We show that it admits an integral representation, in terms of a Hamiltonian that is related to a system of 6m+2 coupled non-linear equations. This system of equations has at least one solution that we derive from the Lax pair of a Riemann-Hilbert problem. We further determine the asymptotics of the generating function, as the size of the intervals become large, up to and including the notoriously difficult constant term. This work generalizes recent results of Dai, Xu and Zhang that correspond to m = 1.