Matrix Integrals,Combinatorics and Integral Lattices

对DMS-0100782(1)随机矩阵的本征值分布是什么，在一定的对称性条件下才能保证谱的真实性？当矩阵的大小变大时，分布会发生什么？极限中的普适性如何？(2)随机排列或随机字中最长递增序列的长度的统计是什么(Ulam的问题)。这个问题适用于界面生长、随机环境中的聚合物、首次通过渗流问题和“二聚体”构型的模型。(3)群和对称空间(或它们的切线空间)上的积分导致了各种有趣的矩阵模型。(微扰)展开式的系数具有显著的组合或拓扑意义，并且可以递归计算。这项工作起源于Feynman，‘t Hooft，Bessis-Itzykson-Zuber和Witten在弦理论的背景下的工作。(4)两个高斯总体的典型相关系数的样本正则相关系数(极大似然估计)是两个总体的统计独立性的检验统计量，如Hotling，James和Constantine所研究的。(5)上述四个问题及其时间扰动都是可积方程或格子的解。对于大型随机矩阵或排列，它们是Korteweg-DeVries方程的解。在有限情况下，它们是Toda格子的解，以及两个新的可积格子，Pfaff和Toeplitz格子的解。除了矩阵积分解，这些格子%有有趣的有理解。公平地说，矩阵积分指向新的可积系统，也指向新的组合和概率问题！一般描述：这些问题涉及到物理、工程和统计学中的一些重要问题：问题(1)起源于对核物理中重核的能级(激发谱)的研究(Wigner，Dyson)。这些能级是如此错综复杂，任何明确的描述都将是困难的。为此，Wigner提出了一个关于这些能级的统计模型。关于问题(2)，众所周知，很大比例的计算机时间用于重新排列计算过程中使用的数据。在一次彻底的洗牌之后，有多少数据在统计上仍然是有序的？这个随机排列的问题也适用于统计力学，热力学的基础，以及聚合物的问题。有些矩阵模型(3)为弦理论提供了玩具模型，弦理论是建立在物理世界基本相互作用基础上的一套重要思想。结果发现，上述某些矩阵模型被用来检验两组统计数据是否相关，如(4)所述。柯特卫格-德弗里斯方程，即拱型‘孤子方程’，描述了波在浅水中的传播；当波长相对于光纤的直径较长时，一个类似的非线性偏微分方程式控制着超短脉冲在光纤中的传播。

Abstract for DMS - 0100782(1) What is the distribution of the eigenvalues of arandom matrix, with certain symmetry conditions toguarantee the reality of the spectrum? What happens tothe distribution, when the size of the matrix getslarge? What about universality in the limit?(2) What is the statistics of the length of thelongest increasing sequence in random permutation orrandom words (Ulam's problem). This question appliesto models of interface growth, polymers in randomenvironments, first passage percolation problem, and "dimer" configurations.(3) Integrals over groups and symmetric spaces, (orover their tangent spaces) lead to a variety ofinteresting matrix models. The coefficients of the(perturbative) expansions have striking combinatorialor topological significance and can be computed in arecursive way. This work originates in the works ofFeynman, 't Hooft, Bessis-Itzykson-Zuber and Witten,in the context of string theory.(4) The sample canonical correlation coefficients(maximum likelihood estimates) for the canonicalcorrelation coefficients of two Gaussian populationsare the test statistic for the statisticalindependence of the two populations, as studied byHotelling, James and Constantine.(5) The four problems above and their"time"-perturbations are all solutions to integrableequations or lattices. For large random matrices orpermutations, they are solutions to the Korteweg-deVries equation. In the finite case, they are solutionsto the Toda lattice, and to two new integrablelattices, the Pfaff and Toeplitz lattices.%Besides matrix integrals solutions, these lattices%have interesting rational solutions.It is fair to say that matrix integrals point the wayto new integrable systems, but also to newcombinatorial and probabilistic questions !General description: The problems aboverelate to a number of important questions in physics,engineering and statistics:Problem (1) has its origin in the study of energylevels (excitation spectra) of heavy nuclei in nuclearphysics (Wigner, Dyson). These levels are so intricatethat any explicit description would be intractable.For that reason, Wigner proposed a statistical modelfor these energy levels. Concerning problem (2), it is well known that a large percentage of computer time is devoted to the rearrangement of the data used in the course of computations. How many data, after a complete reshuffling, are statistically still in order? This question of random permutations also applies tostatistical mechanics, the basis of thermodynamics,and to questions of polymers. Some of the matrix models (3) provide toy models for string theory, an important set of ideas at the basis of the fundamental interactions in the physical world.It turns out that certain matrix models, mentionedabove, are used in testing whether two sets ofstatistical data are correlated, as sketched in (4).The Korteweg-de Vries equation, the arch-type``soliton equation", describes the propagation ofwaves in shallow water; a similar non-linear partialdifferential equation governs the propagation ofultra-short pulses in optical fibers, when the wavelength is long compared to the diameter of the fiber.