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.

该研究项目针对随机矩阵和随机多项式及其应用的基本问题，以及统计物理学的相关问题。问题的核心是普遍性的不同猜想，它表明，作为随机矩阵的大小（或Arandom多项式的程度）接近无穷大，因此，特定缩放的特征值​​（或Zeros）接近普遍限制之间的相关性。 在当前项目中，PI继续他对宇宙矩阵模型，随机多项式和统计物理学的研究。这包括：（i）随机矩阵模型中的Riemann-Hilbert（RH）方法缩放限制。 （ii）与外部源的RH方法Torandom矩阵。 （iii）多矩阵模型的半经典渐近学和RH方法。 （iv）统计物理学的THESIX-VERTEX模型的RH方法。 （v）随机多项式的非高斯合奏中的缩放限制和附加性。该项目具有跨学科的特征，它在物理和数学之间的范围内。在现代科学的许多领域，缩放和非妇女的问题是核心：批评现象和相变理论，统计物理学和Quantum田间理论，量子混乱理论，非线性动力学等。该项目针对范围矩阵，随机多项式和相关主题的规模和普遍性问题的强大数学方法的发展。它涉及数学的不同领域：分析，集成性系统理论，概率理论，半经典渐近学用于分量方程的系统，复杂的分析等。研究项目的考虑直接应用于各种物理问题：组合性渐近性，与量子重力相关的组合渐近性模型物理学，随机曲面上的自旋系统，关键现象和相变的理论，量子。可能的进一步应用包括结和链接理论以及分子生物学，增长模型，统计数据分析等中的相关问题。