Scaling and universality in random matrix models and statistical physics

随机矩阵模型和统计物理中的标度和普适性

• 批准号: 0652005
• 负责人: Pavel Bleher
• 金额: $ 24万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652005&HistoricalAwards=false
• 关键词: Scaling; universality; random; matrix; models

This research project is directed on fundamental problems of thetheory of random matrices and random polynomials and theirapplications, and on related problems in statistical physics. Thecornerstone of the problems is different conjectures of universality,which state that as the size of a random matrix (or the degree of arandom polynomial) approaches infinity, the correlations betweenproperly scaled eigenvalues (or zeros) approach a universal limit. Inthe current project the PI continues his studies of the universalityin random matrix models, random polynomials, and statisticalphysics. This includes: (i) The Riemann-Hilbert (RH) approach todouble scaling limits in random matrix models. (ii) RH approach torandom matrices with external source. (iii) Semiclassical asymptoticsand RH approach to multi-matrix models. (iv) RH approach to thesix-vertex model of statistical physics. (v) Scaling limits anduniversality in non-Gaussian ensembles of random polynomials andrandom algebraic varieties.The project has an interdisciplinary character and it lies on thefrontier between physics and mathematics. The problems of scaling anduniversality are central in many areas of modern science: theory ofcritical phenomena and phase transitions, statistical physics andquantum field theory, theory of quantum chaos, nonlinear dynamics,etc. This project is directed on development of powerful mathematicalmethods to the problems of scaling and universality in the theory ofrandom matrices, random polynomials, and related topics. It involvesdifferent areas of mathematics: analysis, theory of integrablesystems, probability theory, semiclassical asymptotics for systems ofdifferential equations, complex analysis, etc. The research projectunder consideration has direct applications to various physicalproblems: combinatorial asymptotics related to quantum gravity,exactly solvable models of statistical physics, spin systems on randomsurfaces, theory of critical phenomena and phase transitions, quantumchaos. Possible further applications include theory of knots andlinks and related problems in molecular biology, growth models,statistical data analysis, and others.

本研究项目针对随机矩阵和随机多项式理论的基本问题及其应用，以及统计物理中的相关问题。问题的基石是不同的普遍性猜想，即当随机矩阵的大小（或随机多项式的次数）接近无穷大时，适当缩放的特征值​​（或零）之间的相关性接近普遍极限。 在当前的项目中，PI 继续研究随机矩阵模型、随机多项式和统计物理学的普遍性。这包括： (i) 在随机矩阵模型中实现双尺度限制的黎曼-希尔伯特 (RH) 方法。 (ii) RH 使用外部源处理随机矩阵。 (iii) 多矩阵模型的半经典渐进和 RH 方法。 (iv) 统计物理六顶点模型的 RH 方法。 (v) 随机多项式和随机代数簇的非高斯系综的标度极限和普适性。该项目具有跨学科特征，处于物理学和数学之间的前沿。标度和普适性问题是现代科学许多领域的核心：临界现象和相变理论、统计物理和量子场论、量子混沌理论、非线性动力学等。该项目旨在开发强大的数学方法来解决随机矩阵、随机多项式和相关主题理论中的标度和普适性问题。它涉及不同的数学领域：分析、可积系统理论、概率论、微分方程组的半经典渐进、复分析等。正在考虑的研究项目直接应用于各种物理问题：与量子引力相关的组合渐近、统计物理的精确可解模型、随机表面上的自旋系统、临界现象和相变理论、 量子混沌。可能的进一步应用包括结和链接理论以及分子生物学、生长模型、统计数据分析等中的相关问题。