Matrix Integrals,Combinatorics and Integral Lattices

摘要DMS - 0100782（1）随机矩阵的特征值分布是什么，具有一定的对称性条件以保证频谱的真实性？当矩阵变大时，分布会发生什么变化？极限中的普遍性呢？(2)随机排列或随机字中最长递增序列长度的统计量是多少（乌拉姆问题）？这个问题适用于界面生长模型、随机环境中的聚合物模型、首次通过渗流问题和“二聚体”构型。(3)群和对称空间上的积分（或它们的切空间上的积分）导致了各种有趣的矩阵模型。这些（微扰）展开式的系数具有显著的组合或拓扑意义，并且可以用递归的方法计算.这项工作起源于作品ofFeynman，'t Hooft，贝西-Itzykson-Zuber和维滕，在弦理论的背景下。(4)两个高斯总体典型相关系数的样本典型相关系数（极大似然估计）是两个总体独立性的检验统计量，如Hotelling，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。在有限情形下，它们是户田格和两个新的可积格Pfaff格和Toeplitz格的解.除了矩阵积分解之外，这些格还有有趣的有理解。可以说，矩阵积分不仅为新的可积系统指明了方向，而且也为新的组合问题和概率问题指明了方向。一般说明：问题（1）起源于核物理学中重核能级（激发谱）的研究（Wigner，Dyson）。这些能级是如此复杂，任何明确的描述都是困难的。因此，维格纳提出了一个统计模型，这些能级。关于问题（2），众所周知，计算机时间的很大一部分用于重新安排计算过程中使用的数据。有多少数据，在完全重新洗牌后，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。一些矩阵模型（3）为弦论提供了玩具模型，弦论是物理世界中基本相互作用基础上的一组重要思想。事实证明，上面提到的某些矩阵模型用于检验两组统计数据是否相关，如（4）所示。Korteweg-de弗里斯方程，拱形“孤波方程”，描述了波在浅水中的传播;当波长与光纤直径相比较长时，类似的非线性偏微分方程控制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.