Universality and Scaling in Random Matrix Models, Random Polynomials, and Quantum Mechanics and Statistical Physics

随机矩阵模型、随机多项式、量子力学和统计物理中的普适性和标度

• 批准号: 9970625
• 负责人: Pavel Bleher
• 金额: $ 7.33万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970625&HistoricalAwards=false
• 关键词: Universality; Scaling; Random; Matrix; Models

Proposal:DMS-9970625Principal Investigator: Pavel M. BleherAbstract: Bleher's project focuses on basic problems in the theory of random matrices and random polynomials, and on related problems in the theory of quantum chaos and statistical mechanics. The unifying theme of these problems is a number of different conjectures about universality. The major topics of investigation include: (i) universality of the double scaling limit in random matrix models near critical and multicritical points, its relation to integrable hierarchies and Painleve transcendents, and applications of random matrix models; (ii) universality and scaling for random sections of powers of positive line bundles, especially SU(m+1)-polynomials and universality of eigenstates statistics in the theory of quantum chaos; (iii) almost periodicity of the spectral function for integrable systems in quantum mechanics, trace formulae, and probability distribution of the error function; (iv) the Thouless effect and superpolynomial scaling in classical spin systems with long-range interaction.The concept of scaling and universality is a fundamental principle in different theories of modern mathematics and physics, from probability theory to the physical theory of quantum chaos, and from mathematical analysis to the theory of phase transitions and critical phenomena in statistical physics. In rough terms, the concept of universality states that many mathematical and physical phenomena display remarkable universal features on the scale of very small or very large distances. The major goal of the current project is to consolidate the different concepts of universality that arise in various contexts and, in particular, to investigate scaling and universality in mathematical theories of random matrices and random polynomials with a view toward applications to the theory of quantum chaos and to statistical physics. This award is jointly funded by the Analysis Program in the Division of Mathematical Sciences and the Mathematical Physics Program in the Division of Physics.

提案：DMS-9970625原理研究者：Pavel M. Bleabstract：Bleher的项目着重于随机矩阵和随机多项式理论中的基本问题，以及量子混乱和统计力学理论中的相关问题。这些问题的统一主题是关于普遍性的许多不同的猜想。调查的主要主题包括：（i）在关键和多政治点附近的随机矩阵模型中的双缩放限制的通用性，其与可集成的层次结构和Painleve超越者的关系以及随机矩阵模型的应用； （ii）量子混乱理论中特征状态统计量的正线束的幂截面的普遍性和缩放； （iii）在量子力学，痕量公式和误差函数的概率分布中，频谱函数的光谱函数几乎是周期性； （iv）在具有远距离相互作用的经典旋转系统中，您的无效效果和超级合理缩放。缩放和普遍性的概念是现代数学和物理学不同理论的基本原理，从概率理论到量子混乱的物理理论，以及从数学分析到统计物理学中的相变和关键现象的理论。简而言之，普遍性的概念指出，许多数学和物理现象在非常小或非常大的距离的规模上表现出了非凡的普遍特征。当前项目的主要目的是巩固在各种情况下出现的不同普遍性的不同概念，尤其是研究随机矩阵的数学理论和随机多项式的数学理论的扩展和普遍性，并认为适用于量子混乱理论和统计物理学。该奖项由数学科学系和物理学部数学物理计划的分析计划共同资助。