Universality in non-Hermitian matrix models

非厄米矩阵模型中的普遍性

• 批准号: EP/G019843/1
• 负责人: Francesco Mezzadri
• 金额: $ 61.92万
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2009
• 资助国家: 英国
• 起止时间: 2009 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FG019843%2F1
• 关键词: Universality; non; Hermitian; matrix; models

Universality is a fundamental aspect of our understanding of Nature. It means that many physical systems manifest the same behaviour independently of what the details of the interaction among their constituent elements are. For example, all macroscopic objects obey the laws of thermodynamics, while at the same time matter is built out of atoms. The conciliation of the macroscopic laws of thermodynamics with atomic physics has been a long-standing, fundamental and difficult challenge for scientists. In other words, on a macroscopic scale physical systems exhibit universality.Loosely speaking, Random Matrix Theory (RMT) can be thought of as a combination of linear algebra and the theory of probability. Each time a physical or mathematical process has a stochastic nature and its governed by linear equations, it is likely that it may be modelled by RMT. Indeed, random matrix models have fundamentally important applications in many branches of mathematics and physics such as combinatorics, complex systems, dynamical systems, growth problems, integrable systems, number theory, operator algebra, probability theory, quantum chaos, quantum field theory, quantum information, statistics, statistical mechanics, structural dynamics and telecommunications. The main feature that makes RMT a powerful tool in such wide range of applications is, once again, universality. In this context it means that for large matrix dimensions the local statistics of eigenvalues of random matrices depend only on the symmetries of the matrices, but are independent of the choice of the probability densities that govern their stochastic behaviour. Universality has been proved for a large class of Hermitian matrix models, but in its full generality it is still a conjecture.The main goal of this project is to prove universality in a vast class of non-Hermitian matrix models. Such ensembles of matrices find applications to growth problems, to the Hele-Shaw problem, especially in the vicinity of a critical point, and to semiclassical study of electronic droplets in the Quantum Hall regime. Non-Hermitian matrices do not have any symmetry constraints, with the exception that their elements must be real, complex or real quaternions respectively. The universality of the spectra of random matrices will be studied in all these three cases. Furthermore, universality will be investigated in the bulk as well in singular regions of the spectrum. It is expected that in these two cases the local spectral statistics will behave rather differently. However, they will still be universal, in the sense that they will depend only on the type of critical point but not on the probability distribution of the matrices.Finally, one of the main tools in the investigation of universality in non-Hermitian matrix models will be the asymptotic analysis of orthogonal polynomials in the complex plain using the dbar problem, which will have implications, for example, in the study of dispersionless multi-dimensional integrable systems and in the asymptotic analysis of integrable operators.

普遍性是我们理解自然的一个基本方面。这意味着许多物理系统表现出相同的行为，而不受其组成元素之间相互作用的细节的影响。例如，所有宏观物体都遵守热力学定律，而同时物质是由原子构成的。热力学宏观定律与原子物理的协调一直是科学家面临的一个长期的、基本的和困难的挑战。换句话说，在宏观尺度上，物理系统表现出普适性。更确切地说，随机矩阵理论(RMT)可以被认为是线性代数和概率论的结合。每当一个物理或数学过程具有随机性并由线性方程控制时，它很可能是由RMT建模的。事实上，随机矩阵模型在许多数学和物理分支中都有重要的应用，如组合学、复杂系统、动力系统、增长问题、可积系统、数论、算符代数、概率论、量子混沌、量子场论、量子信息、统计学、统计力学、结构动力学和电信。使RMT在如此广泛的应用中成为一个强大工具的主要特征再次是通用性。在这种情况下，这意味着对于大的矩阵维度，随机矩阵的特征值的局部统计量仅取决于矩阵的对称性，而与支配其随机行为的概率密度的选择无关。虽然已经证明了一大类厄米特矩阵模型的普适性，但在它的全部普适性方面，它仍然是一个猜想。本项目的主要目的是证明一大类非厄米特矩阵模型的普适性。这种矩阵系综可应用于生长问题、Hele-Shaw问题，特别是在临界点附近，以及量子霍尔区中电子液滴的半经典研究。非厄米特矩阵没有任何对称约束，只是它们的元素必须分别是实数、复数或实四元数。在这三种情况下，我们将研究随机矩阵的谱的普适性。此外，普适性也将在光谱的奇异区域中进行整体研究。预计在这两种情况下，局域光谱统计将表现得相当不同。最后，研究非厄米特矩阵模型的普适性的主要工具之一将是利用dbar问题对复平面上的正交多项式进行渐近分析，这将在研究无色散多维可积系统和可积算子的渐近分析中产生影响。

项目成果

The Hermitian two matrix model with an even quartic potential

具有偶四次势的埃尔米特二矩阵模型

• 发表时间: 2012
• 期刊: MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY
• 影响因子: 1.9
• 作者: Duits Maurice

On Relations between One-Dimensional Quantum and Two-Dimensional Classical Spin Systems

一维量子与二维经典自旋系统的关系

• DOI: 10.1155/2015/652026
• 发表时间: 2015
• 期刊: Advances in Mathematical Physics
• 影响因子: 1.2
• 作者: Hutchinson J

Random matrix theory and critical phenomena in quantum spin chains.

• DOI: 10.1103/physreve.92.032106
• 发表时间: 2015-03
• 期刊: Physical review. E, Statistical, nonlinear, and soft matter physics
• 作者: J. Hutchinson;Jon P Keating;F. Mezzadri

Rate of convergence of linear functions on the unitary group

酉群上线性函数的收敛率

• DOI: 10.48550/arxiv.1009.0695
• 发表时间: 2010
• 作者: Keating J

Random matrix theory and critical phenomena in quantum spin chains

随机矩阵理论和量子自旋链中的临界现象

• DOI: 10.48550/arxiv.1503.05732
• 发表时间: 2015
• 作者: Hutchinson J