Exact and asymptotic methods in Random Matrix Theory and Integrable Systems

随机矩阵模型是由Wigner引入的，用于模拟核相互作用，后来由于与理论物理(弦理论和二维量子重力)、纯数学和应用数学(数论、组合学、金融统计建模)、不可压缩粘性介质流体动力学(Hele-Shaw Cells)以及无限维完全可积动力系统理论等领域的广泛联系，重新引起了人们的兴趣。正在进行的研究项目专注于随机矩阵理论(RMT)与某些已知的动力学系统之间的关系，这些系统被命名为Toda，Volterra，相对论Toda，Ablwitz-Ladik，2-Toda晶格等。这个项目的一个小部分致力于与所谓的不可压缩流体的拉普拉斯增长问题的关系。在随机矩阵理论中，主要的兴趣是描述矩阵(特征值)的谱的统计性质，特别是当矩阵的大小时，这些特征值之间的概率密度和所有相关函数可以用某些多项式(正交多项式及其变体，如双正交多项式)来表示。这些统计函数在核物理最初动机中的重要性在于，本征值代表量子系统的能量，知道它们的分布可以在统计的基础上预测此类核样本的发射和吸收光谱。过去三年的主要发现之一是与这些多项式自然相关的“谱曲线”的相关性。这个项目的主要目标之一是将频谱曲线技术扩展到越来越大的RMT区域，并使用代数结构来推断否则无法获得的非微扰渐近信息(例如-渐近频谱曲线的最大亏格等)。

Random matrix models were introduced by Wigner to model nuclear interactions and have witnessed a renewed interest later on due to unexpected (at the time) ties with a widespread range of areas in both theoretical physics (string theory and two-dimensional quantum gravity), pure and applied mathematics (number theory, combinatorics, statistical modeling of finance) fluid dynamics of incompressible viscous media (Hele-Shaw cells)  and the theory of completely integrable dynamical systems in infinite dimensions. The ongoing project of research focuses on this latter  relationship between random matrix theory (RMT)  and certain dynamical systems known under the names of Toda, Volterra, Relativistic Toda, Ablowitz-Ladik, 2-Toda lattices etc. A minor component of this projects is devoted to the relation with the so-called Laplacian growth problems of incompressible fluids. In random matrix theory the main interest is to describe the statistical properties of the spectrum of the matrices (the eigenvalues), in particular when the size of the matrix is large.  The probability densities and all correlation functions between these eigenvalues can be expressed in terms of certain polynomials (orthogonal polynomials and variations thereof like biorthogonal polynomials). The importance of these statistical functions in the original motivation for nuclear physics was that an eigenvalue represent an energy of the quantum system and knowing their distributions allows to predict on a statistical basis the spectra in emission and absorption of samples of such nuclei. One of the main findings of the past three years has been the relevance of a "spectral curve" associated naturally to these polynomials. It is one of the main aims of this project to extend the spectral curve-technique to increasingly larger areas of RMT and to use the algebraic structure to infer otherwise unaccessible  nonperturbative asymptotic informations (such as -e.g- the maximal genus of the asymptotic spectral curve etc.)