Program in Renormalization and Universality in Mathematics and Mathematical Physics

摘要本课题主要研究随机矩阵和随机多项式理论的基本问题及其应用，以及统计物理中的相关问题。这些问题的基础是不同的普适性猜想，即当随机矩阵的大小（或随机多项式的程度）接近无穷大时，适当缩放的特征值（或零）之间的相关性接近普适性极限。在目前的项目中，PI继续研究随机矩阵模型，随机多项式和统计物理的通用性。这包括：(i) Riemann-Hilbert （RH）方法在随机矩阵模型中的双尺度极限。（ii） RH方法对具有外部源的随机矩阵。（iii）多矩阵模型的半经典渐近性和RH方法。（iv）统计物理六顶点模型的RH方法。(v)随机多项式和随机代数变的非高斯系的标度极限和普适性。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、随机多项式和相关主题中的标度和普适性问题。它涉及数学的不同领域：分析、可积系统理论、概率论、微分方程系统的半经典渐近、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。

ABSTRACTBleherThis research project is directed on fundamental problems of thetheory of random matrices and random polynomials and theirapplications, and on related problems in statistical physics. Thecornerstone of the problems is different conjectures of universality,which state that as the size of a random matrix (or the degree of arandom polynomial) approaches infinity, the correlations betweenproperly scaled eigenvalues (or zeros) approach a universal limit. Inthe current project the PI continues his studies of the universalityin random matrix models, random polynomials, and statisticalphysics. This includes: (i) The Riemann-Hilbert (RH) approach todouble scaling limits in random matrix models. (ii) RH approach torandom matrices with external source. (iii) Semiclassical asymptoticsand RH approach to multi-matrix models. (iv) RH approach to thesix-vertex model of statistical physics. (v) Scaling limits anduniversality in non-Gaussian ensembles of random polynomials andrandom algebraic varieties.The project has an interdisciplinary character and it lies on thefrontier between physics and mathematics. The problems of scaling anduniversality are central in many areas of modern science: theory ofcritical phenomena and phase transitions, statistical physics andquantum field theory, theory of quantum chaos, nonlinear dynamics,etc. This project is directed on development of powerful mathematicalmethods to the problems of scaling and universality in the theory ofrandom matrices, random polynomials, and related topics. It involvesdifferent areas of mathematics: analysis, theory of integrablesystems, probability theory, semiclassical asymptotics for systems ofdifferential equations, complex analysis, etc. The research projectunder consideration has direct applications to various physicalproblems: combinatorial asymptotics related to quantum gravity,exactly solvable models of statistical physics, spin systems on randomsurfaces, theory of critical phenomena and phase transitions, quantumchaos. Possible further applications include theory of knots andlinks and related problems in molecular biology, growth models,statistical data analysis, and others.