Random matrix models and their applications to statistical mechanics

本课题的研究方向是随机矩阵模型理论的基本问题及其在统计力学中的应用。该项目涉及特征值相关函数标度极限的普适性的不同猜想和统计物理的精确可解模型。这包括：(a)发展带有外部源的随机矩阵模型的Riemann-Hilbert （RH）方法；(b)多矩阵模型的半经典渐近性和RH方法；(c)带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，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。该项目旨在开发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、复分析等。正在考虑的研究项目可直接应用于各种物理问题：与量子引力相关的组合渐近性、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、生长模型、统计数据分析等方面的结和链接理论及相关问题。

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.