Random matrices and integrable systems

这个程序涉及可积系统理论和随机矩阵谱理论之间的接口。这项工作的动机来自于随机矩阵理论的结果在物理学的不同领域的非常广泛的适用性。其中最突出的是：它的关系增长现象，确定性和随机性;计算相关函数的超级杨米尔斯理论;问题的图形枚举，二维量子引力，和共形模型。 所使用的工具主要来自于可积系统研究中发展起来的逆谱方法。这两个主题之间的关系在于，出现在定义随机矩阵系综的测度中的参数和那些确定支撑集的参数可以被视为动态变形参数。随机矩阵理论中研究的统计量满足关于这些参数的偏微分和差分方程系统，这些参数恰好是进入可积系统理论的类型。 这个程序的一个主要目标是扩展到多矩阵设置的非线性渐近方法，已经成功地应用于研究大N渐近的一个矩阵的情况下。我们能够将这种方法扩展到多矩阵的情况下，因为基本的双正交多项式的黎曼-希尔伯特特征是已知的，所以是“模型解决方案”，在通常的分析工具中使用的函数和变形理论的黎曼曲面。 另一种正在开发的方法是基于自由费米子量子场，源于京都学派关于可积层次的早期工作。该方法已成功地用于计算二元矩阵和复矩阵积分的特征展开式，以及特征多项式的有理相关函数。这些结果应承认扩展到更一般的矩阵群，以及多矩阵模型。

This program concerns the interface between the theory of integrable systems and the spectral theory of random matrices. The motivation for this work derives from the very wide applicability of results from random matrix theory in diverse areas of physics. Amongst the most prominent are: its relation to growth phenomena, both deterministic and stochastic; the computation of correlation functions in super Yang-Mills theory; problems of graphical enumeration, two dimensional quantum gravity, and conformal models.     The tools used are drawn mainly from the inverse spectral methods developed in the study of integrable systems. The relation between the two topics lies in the fact that parameters appearing in the measure defining random matrix ensembles and those determining the support set may be viewed as dynamical deformation parameters. The statistical quantities studied in random matrix theory satisfy systems of partial differential and difference equation with respect to these parameters that turn out to be exactly of the type entering in the theory of integrable systems.    One main objective of this program is to extend to the multi-matrix setting the nonlinear asymptotic methods that have been so successfully applied in studying the large N asymptotics of the one-matrix case. We are in a position to extend this approach to the multi-matrix case since the Riemann-Hilbert characterization of the underlying biorthogonal polynomials is known, and so is the "model solution", in terms of the usual analytic tools used in the function and deformation theory on Riemann surfaces.   Another approach that is being developed is based on free fermion quantum fields, stemming from the early work of the Kyoto school on integrable hierarchies. This method has been successfully used so far for computing character expansions of two-matrix and complex matrix integrals, as well as rational correlations functions of characteristic polynomials. These results should admit extensions to more general matrix groups, as well as to multi-matrix models.