Random matrix models and their applications

本研究项目是针对随机矩阵模型理论及其在统计物理中的应用的基本问题进行的。该项目涉及对特征值关联函数标度极限普适性的不同猜想，以及统计物理和流体力学的精确可解模型。这包括：(A)随机正态矩阵模型的Riemann-Hilbert方法的发展及其在流体力学和拉普拉斯增长中的Hele-Shaw流动的应用；(B)Riemann-Hilbert方法对具有外部源的随机矩阵模型的发展；(C)铁电和反铁电相中带磁壁边界条件的六顶点模型的精确解；(D)六顶点模型中的相分离的研究和其他。该项目具有跨学科的特点，它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核心问题：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。这个项目发展了强大的数学方法来解决随机矩阵理论及其应用中的标度和普适性问题。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。该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。

This research project is directed at basic problems of the theory of random matrix models and their applications to statistical physics. The project concerns with different conjectures of universality of the scaling limits of eigenvalue correlation functions and with 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 and Laplacian growth; (b) the development of the Riemann-Hilbert approach to random matrix models with external source; (c) the exact solution of the six-vertex model with domain wall boundary conditions in ferroelectric and antiferroelectric phases; (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 develops powerful mathematical methods to solve the problems of scaling and universality in the theory of random matrices and their applications. It involves methods of different areas of mathematics: analysis, theory of integrable systems, probability theory, semiclassical asymptotics, complex analysis, etc. The research project 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.