Random Matrix Models and Statistical Mechanics

本研究项目在物理和数学的前沿具有跨学科的特点。该项目研究了随机矩阵中的尺度和普适性现象，这是现代科学许多领域的核心。提高对这些现象的理解将对数学理论及其应用产生重大影响。该项目旨在开发新的、强大的数学方法，涉及不同数学领域的方法，包括分析、可积系统理论、概率论、半经典渐近和复分析。该项目的结果将直接应用于量子引力的组合方面，统计力学的精确可解模型，随机表面上的自旋系统，临界现象和相变理论，以及量子混沌。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链接理论和相关问题。这个研究项目是针对随机矩阵模型理论和统计力学的基本问题。本项目研究了统计物理和流体力学中特征值相关函数标度极限的普适性和精确可解模型的不同猜想。这包括：(a)发展随机正态矩阵模型的Riemann-Hilbert方法，并将其应用于流体力学中的Hele-Shaw流、拉普拉斯增长和量子霍尔效应；(b)发展了具有外部源的随机矩阵模型的Riemann-Hilbert方法，包括这些模型中的相变、临界现象和双标度极限问题；(c)不同边界条件下六顶点模型在不同阶段的精确解；(d)六顶点模型中相分离的研究。

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.