Random matrix models and their applications to statistical mechanics

随机矩阵模型及其在统计力学中的应用

• 批准号: 0969254
• 负责人: Pavel Bleher
• 金额: $ 16.5万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0969254&HistoricalAwards=false
• 关键词: Random; matrix; models; their; applications

This research project is directed on basic problems of the theory of random matrix models and their applications to statistical mechanics. The project concerns with different conjectures of universality of the scaling limits of eigenvalue correlation functions and with exactly solvable models of statistical physics. This includes: (a) the development of the Riemann-Hilbert (RH) approach to random matrix models with external source; (b) the semiclassical asymptotics and the RH approach to multi-matrix models; (c) the exact solution of the six-vertex model with domain wall boundary; (d) the study of the phase separation in the six-vertex model, and others.The project has an interdisciplinary character and it lies on the frontier between physics and mathematics. The problems of scaling and universality are central in many areas of modern science: theory of critical phenomena and phase transitions, statistical physics and quantum field theory, theory of quantum chaos, nonlinear dynamics, etc. This project is directed on the development of powerful mathematical methods to the problems of scaling and universality in the theory of random matrices and their applications to statistical physics. It involves methods of different areas of mathematics: analysis, theory of integrable systems, probability theory, semiclassical asymptotics, complex analysis, etc. The research project under consideration has direct applications to various physical problems: combinatorial asymptotics related to quantum gravity, exactly solvable models of statistical physics, spin systems on random surfaces, theory of critical phenomena and phase transitions, quantum chaos. Possible further applications include the theory of knots and links and related problems in molecular biology, growth models, statistical data analysis, and others.

本研究课题是关于随机矩阵模型理论及其在统计力学中应用的基本问题的研究。该项目涉及对本征值关联函数标度极限普适性的不同猜想，以及统计物理的精确可解模型。这包括：(A)将Riemann-Hilbert(RH)方法推广到具有外部源的随机矩阵模型；(B)半经典渐近性和RH方法用于多矩阵模型；(C)具有域壁边界的六顶点模型的精确解；(D)六顶点模型中相分离的研究等。该项目具有跨学科的特点，位于物理和数学之间的前沿。标度和普适性问题是现代科学许多领域的中心问题：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。本项目旨在发展强大的数学方法来解决随机矩阵理论中的标度和普适性问题及其在统计物理中的应用。它涉及不同数学领域的方法：分析、可积系统理论、概率论、半经典渐近性、复分析等。正在考虑的研究项目直接应用于各种物理问题：与量子引力有关的组合渐近性、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分子生物学中的结链理论和相关问题、生长模型、统计数据分析等。