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.

我的研究计划涉及算子理论和复分析及其在随机矩阵理论和数学物理中的应用。更准确地说，我的目标是关注以下三个领域：（A）Toeplitz行列式与Fisher-Hartwig（F-H）奇点和Riemann-Hilbert问题的双尺度限制； (B) 软黎曼-希尔伯特问题，它建立在 Hedenmalm 和 Wenmann 最近突破的基础上； (C) 广义希尔伯特矩阵的谱特性。在主题 A 中，我的目标是在存在有限多个 F-H 奇点的情况下计算托普利茨行列式的双标度极限，当行列式的大小趋于无穷大并且外部参数趋于临界时，其中至少两个会合并为一个。同时值。我利用黎曼-希尔伯特问题（RHP）方法和算子理论技术来研究这个问题。我还考虑了托普利茨行列式的双标度极限在随机矩阵理论中的应用。在主题 B 中，我研究了二维行列式点过程模型中出现的软 RHP，例如随机正态矩阵，其中特征值很复杂，并且倾向于填充二维正区域集。与经典的 RHP 相比，软 RHP 的研究还处于起步阶段，我的目标是进一步发展其理论并将其应用于可积非线性 PDE 问题。在主题 C 中，我的主要目标是研究解析函数空间上广义希尔伯特矩阵算子的有界性和谱性质。特别是，我的目标是通过在 Hardy 空间、序列空间和 Korenblum 空间中描述这些算子来完成这些算子的谱图。另一个目标是开发用于表征作用于解析函数的 Banach 空间的一大类 Hankel 算子的谱的技术以及这些空间的一般方法。