Fluctuations on Global and Intermediate Scales for Orthogonal Polynomial Ensembles
Time: Thu 2025-02-20 10.00
Location: F3 (Flodis), Lindstedtsvägen 26 & 28, Stockholm
Video link: https://kth-se.zoom.us/j/61010774094
Language: English
Doctoral student: Wenkui Liu , Sannolikhetsteori, matematisk fysik och statistik
Opponent: Professor Igor Krasovsky, Imperial College London
Supervisor: Maurice Duits, Matematik (Avd.)
QC 2025-02-06
Abstract
Orthogonal polynomial ensembles (OPEs) arise naturally in many models of statistical mechanics, probability theory, combinatorics, and random matrix theory. An important and well-known source of examples of OPEs is the eigenvalues of random Hermitian matrices. Random matrix theory began in 1928 with John Wishart's work, aimed at analysing large datasets. Over the past several decades, the mathematical theory has seen significant advancements, and it continues to be a vibrant area of research today.
Another motivation for studying OPEs comes from random tilings. Random tiling models exhibit phase transitions, where different regions of a tiling exhibit distinct behaviour (e.g., frozen vs. liquid regions). This resembles physical systems with phase boundaries, making random tiling a simple yet powerful model to study phenomena such as crystallisation.
This thesis is a synthesis of two original studies of the asymptotic behaviour of OPEs. The first part of the thesis consists of an introduction, an overview of the papers, and an outlook of the topics studied. The second part of the thesis includes the original papers.
Paper A studies the lozenge tilings of a hexagon with the $q$-Racah weights, which serve as a generalisation of the uniform and $q$-volume weights. We show that the height function for this model concentrates near a deterministic limit shape and that the global fluctuations are described by the Gaussian Free Field (GFF).
Paper B is about OPEs at the mesoscopic scales, which is an intermediate scale between global and local ones. We show that a large class of OPEs have the same limiting fluctuations at the edges in the mesoscopic regime. We extend the method of Breuer and Duits (2016) to varying weights. This approach does not make smoothness assumptions on weights. Hence, our results apply to both continuous and discrete OPEs.