Random matrix models and their applications

本研究计画针对随机矩阵模型理论的基本问题及其在统计物理上的应用。该项目涉及本征值相关函数的标度极限的普适性的不同构造，以及统计物理和流体力学的精确可解模型。这包括：（a）随机正规矩阵模型的Riemann-Hilbert方法的发展及其在流体力学和Laplacian增长中Hele-Shaw流的应用;（B）具有外部源的随机矩阵模型的Riemann-Hilbert方法的发展;（c）铁电和反铁电相中具有畴壁边界条件的六顶点模型的精确解;（d）研究六顶点模型中的相分离等。该项目具有跨学科性质，处于物理学和数学的前沿。标度和普遍性的问题是现代科学的许多领域的中心：临界现象和相变理论，统计物理学和量子场论，量子混沌理论，非线性动力学等。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，复分析等研究项目直接应用于各种物理问题：与量子引力相关的组合渐近，统计物理学的精确可解模型，随机表面上的自旋系统，临界现象和相变理论，量子混沌。可能的进一步应用包括结和链接理论以及分子生物学，生长模型，统计数据分析等相关问题。

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.