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

摘要：Bleher的课题主要研究随机矩阵和随机多项式理论中的基本问题，以及量子混沌理论和统计力学中的相关问题。这些问题的统一主题是一些关于普遍性的不同猜想。主要研究课题包括：(1)临界点和多临界点附近随机矩阵模型的双标度极限的普适性，它与可积层次和Painleve超越的关系，以及随机矩阵模型的应用；（ii）量子混沌理论中正线束幂随机部分的普适性和标度，特别是SU(m+1)-多项式和本征态统计的普适性；(3)量子力学中可积系统谱函数的概周期性、迹式和误差函数的概率分布；(4)具有远距离相互作用的经典自旋系统的Thouless效应和超多项式标度。从概率论到量子混沌的物理理论，从数学分析到统计物理中的相变和临界现象理论，标度和普适性的概念是现代数学和物理不同理论中的一个基本原理。概括地说，普适性的概念是指许多数学和物理现象在非常小或非常大的距离范围内表现出显著的普适性。当前项目的主要目标是巩固在各种情况下出现的不同的普适性概念，特别是研究随机矩阵和随机多项式数学理论中的尺度和普适性，以期应用于量子混沌理论和统计物理。该奖项由数学科学部的分析项目和物理部的数学物理项目共同资助。

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.