Asymptotics of Toeplitz determinants, soft Riemann-Hilbert problems and generalised Hilbert matrices (HilbertToeplitz)

Toeplitz 行列式的渐进性、软黎曼-希尔伯特问题和广义希尔伯特矩阵 (HilbertToeplitz)

基本信息

批准号：
EP/X024555/1
负责人：
Jani A. Virtanen
金额：
$ 24.26万
依托单位：
University of Reading
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2023
资助国家：
英国
起止时间：
2023 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FX024555%2F1
关键词：
Asymptotics Toeplitz determinants soft Riemann

项目摘要

My research proposal is concerned with topics in operator theory and complex analysis with applications to random matrix theory and mathematical physics. More precisely, I aim to focus on the following three areas: (A) double-scaling limits of Toeplitz determinants with Fisher-Hartwig (F-H) singularities and Riemann-Hilbert problems; (B) soft Riemann-Hilbert problems, which builds on a recent breakthrough of Hedenmalm and Wenmann; and (C) spectral properties of generalised Hilbert matrices. In theme A, I aim to compute the double-scaling limits of Toeplitz determinants in the presence of finitely many F-H singularities when at least two of them merge into one as the size of the determinants tends to infinity and an external parameter tends to a critical value simultaneously. I utilise Riemann-Hilbert problem (RHP) method and operator-theoretic techniques to study this problem. I also consider the applications of double-scaling limits of Toeplitz determinants in random matrix theory. In theme B, I investigate Soft RHPs that arise in two-dimensional determinantal point process models, such as the Random Normal Matrix, where the eigenvalues are complex, and tend to fill a two-dimensional set of positive area. Compared with the classical RHPs, the study of soft RHPs is at its infancy and my goal is to develop their theory further and apply it to problems in integrable nonlinear PDEs. In theme C, my main goal is to study boundedness and spectral properties of generalised Hilbert matrix operators on analytic function spaces. In particular, I aim to complete the spectral picture of these operators by describing it in the Hardy spaces, sequence spaces and Korenblum spaces. A goal is also to develop techniques to be used to characterise the spectra of a large class of Hankel operators acting on Banach spaces of analytic functions and methods for these spaces in general.

我的研究建议涉及运营商理论和复杂分析的主题，并应用了随机矩阵理论和数学物理学的应用。更确切地说，我的目标是集中于以下三个领域：（a）Fisher-Hartwig（F-H）奇异性和Riemann-Hilbert问题的Toeplitz决定因素的双重限制；（b）柔软的Riemann-Hilbert问题，基于Hedenmalm和Wenmann的最新突破；（c）广义希尔伯特矩阵的光谱特性。在主题A中，我的目标是在有限的许多F-H奇点存在下计算Toeplitz决定因素的双尺度限制，而其中两个奇异性至少合并为一个，因为决定因素的大小倾向于无限，并且外部参数趋于关键参数。同时价值。我利用Riemann-Hilbert问题（RHP）方法和操作者理论技术来研究此问题。我还考虑了Toeplitz决定因素在随机矩阵理论中的双尺度限制的应用。在主题B中，我研究了在二维确定点过程模型中出现的软RHP，例如随机正常矩阵，其中特征值很复杂，并且倾向于填充二维正面面积。与经典RHP相比，软RHP的研究仍处于起步阶段，我的目标是进一步发展其理论并将其应用于可集成的非线性PDE中的问题。在主题C中，我的主要目标是研究广义希尔伯特矩阵操作员在分析功能空间上的界限和光谱特性。特别是，我旨在通过在耐寒的空间，序列空间和科伦布鲁姆空间中描述这些操作员的光谱图片。一个目标还是开发技术，以表征一大类汉克尔操作员的光谱，这些汉克尔操作员一般而言这些空间的分析功能和方法。