Riemann-Hilbert Problems, Toeplitz Determinants and Applications

数学、物理和工程中的许多问题都可以用Toeplitz矩阵（与主对角线平行的对角线都是常数的矩阵）来表示。近年来，利用Riemann-Hilbert分析和算子理论方法对Toeplitz行列式的渐近研究在随机矩阵理论和数学物理中得到了重要的应用。特别有趣的是由光滑函数生成的Toeplitz行列式的Szegö渐近性和由可能具有不连续、零、卷绕和（可积）奇点的函数生成的Fisher-Hartwig渐近性。具有外部参数T的Toeplitz行列式的渐近研究证明了双标度极限在数学和物理中的重要性。例如，对于二维Ising模型和XY自旋链模型，Toeplitz行列式可以用来分析某些相关函数在T趋于临界值Tc时的双标度极限，同时自旋之间的距离趋于无穷大。作为具体应用，该提案包括对纠缠的研究，纠缠是一种量子现象，为量子计算和信息处理提供了主要资源。作为“量子”的基本度量，它显示了我们可以观察到多少量子效应，并利用多少量子效应来控制一个子系统。在过去的二十年里，人们已经做出了大量的努力来理解各种量子系统中的纠缠。特别令人感兴趣的是所谓的XY量子自旋链，它可以被视为量子计算机的玩具模型。该模型由排列成直线的粒子组成，并以仅具有最近邻相互作用的正整数为索引。由于XY自旋链允许对物理相关量进行精确计算，因此它们为现实生活中的物理系统提供了有价值的见解，而这种计算目前是不可能的。提出的研究方法包括黎曼-希尔伯特问题（RHP），该问题具有悠久而令人印象深刻的历史，可以追溯到黎曼的论文（1851）和希尔伯特在20世纪初的相关结果。黎曼-希尔伯特问题可以被描述为在复平面上寻找一个解析函数的问题，它在给定曲线上有一个规定的跳跃，它与一维奇异积分算子、卷积算子、Toeplitz算子和Wiener-Hopf算子密切相关。除了黎曼-希尔伯特方法，新的算子理论方法将发展与具体算子和矩阵有关。

A considerable variety of problems in mathematics, physics and engineering can be expressed in terms of Toeplitz matrices (matrices whose diagonals parallel to the main diagonal are all constants). Recently the asymptotic study of Toeplitz determinants has found important applications in random matrix theory and mathematical physics using the Riemann-Hilbert analysis and operator theoretic methods. Of particular interest are the so-called Szegö asymptotics of Toeplitz determinants generated by smooth functions and Fisher-Hartwig asymptotics generated by functions that may possess discontinuities, zeros, winding and (integrable) singularities.The asymptotic study of Toeplitz determinants with an external parameter T demonstrates the importance of double-scaling limits in mathematics and physics. For example, regarding the 2D Ising model and the XY spin chain model, the Toeplitz determinants can be used to analyze the double-scaling limit for certain correlation functions as T goes to the critical value Tc and simultaneously the separation between the spins goes to infinity.As a concrete application, the proposal includes the study of entanglement, which is a quantum phenomenon providing a primary resource for quantum computation and information processing. As a fundamental measure of "quantumness," it shows how much quantum effect we can observe and use to control one subsystem by another. In the last two decades, substantial effort has been made to understand entanglement in various quantum systems. Of particular interest is the so-called XY quantum spin chain, which can be viewed as a toy model for quantum computers. The model consists of particles ordered in a line and indexed by the positive integers with only nearest neighbor interactions. Because the XY spin chain allows for exact calculations for physically relevant quantities, they provide valuable insight into real-life physical systems where such calculations are currently impossible.The methods of the proposed research include the Riemann-Hilbert problem (RHP), which has a long and impressive history going back to Riemann's dissertation (1851) and Hilbert's related results at the beginning of the 20th century. The Riemann-Hilbert problem, which can be described as a problem of finding an analytic function in the complex plane with a prescribed jump across a given curve, is closely connected to one-dimensional singular integral operators, convolution operators, Toeplitz operators, and Wiener-Hopf operators. In addition to the Riemann-Hilbert approach, new operator-theoretic methods will be developed related to concrete operators and matrices.