Random Matrix Models and Statistical Mechanics

随机矩阵模型和统计力学

• 批准号: 1565602
• 负责人: Pavel Bleher
• 金额: $ 29.92万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1565602&HistoricalAwards=false
• 关键词: Random; Matrix; Models; Statistical; Mechanics

This research project has an interdisciplinary character on the frontier between physics and mathematics. The project investigates the phenomena of scaling and universality in random matrices, which are central in many areas of modern science. Improved understanding of these phenomena will have a significant impact on both mathematical theory and its applications. The project aims to develop new and powerful mathematical approaches involving methods from different areas of mathematics including analysis, the theory of integrable systems, probability theory, semiclassical asymptotics, and complex analysis. The results of the project will have direct applications to combinatorial aspects of quantum gravity, exactly solvable models of statistical mechanics, spin systems on random surfaces, theory of critical phenomena and phase transitions, and quantum chaos. Further applications include the theory of knots and links and related problems in molecular biology, growth models, and statistical data analysis. This research project is directed at fundamental problems of the theory of random matrix models and statistical mechanics. The project investigates different conjectures of universality of the scaling limits of eigenvalue correlation functions and exactly solvable models of statistical physics and hydrodynamics. This includes: (a) the development of the Riemann-Hilbert approach to random normal matrix models with applications to Hele-Shaw flows in hydrodynamics, Laplacian growth, and quantum Hall effect; (b) the development of the Riemann-Hilbert approach to random matrix models with external source, including the problems of phase transitions, critical phenomena, and double scaling limits in these models; (c) the exact solution of the six-vertex model with various boundary conditions in different phases; and (d) the study of the phase separation in the six-vertex model.

该研究项目在物理和数学之间的前沿具有跨学科的特征。 该项目调查了随机矩阵中缩放和普遍性的现象，在现代科学的许多领域都是核心。 对这些现象的改进理解将对数学理论及其应用产生重大影响。该项目旨在开发涉及来自数学不同领域的方法，包括分析，综合系统理论，概率理论，半经典渐近学和复杂分析的方法。该项目的结果将直接应用于量子重力的组合，统计力学的确切可解决模型，随机表面上的自旋系统，临界现象和相变理论以及量子混乱。进一步的应用包括分子生物学，生长模型和统计数据分析中的结和联系理论以及相关问题。该研究项目针对随机矩阵模型和统计力学理论的基本问题。该项目研究了特征值相关函数的缩放限制范围的不同猜想以及统计物理和流体动力学的准确可解决模型。这包括：（a）开发Riemann-Hilbert方法，用于随机正常矩阵模型，并在水动力学，Laplacian生长和量子Hall效应中应用用于Hele-Shaw流动； （b）开发带有外部源的随机矩阵模型的Riemann-Hilbert方法，包括这些模型中相变，关键现象和双缩放限制的问题； （c）具有不同阶段的各种边界条件的六个vertex模型的精确解； （d）在六个vertex模型中对相分离的研究。