典型群上的积分，随机置换，Toda和Toeplitz格
Integrals over classical Groups, Random permutations, Toda and Toeplitz lattices
摘要：
典型群O+（n），O-（n），Sp（n）和U（n）上的矩阵类Fourier-like积分与随机置换和随机对合中最长递增序列的长度分布以及随机矩阵的谱有关.本文的目的之一是证明所有这些积分满足具有特定初始条件的Painlevé V方程。在这项工作中，我们提出，新的结果和已知的，在一个统一的方式。我们的方法包括插入一组时间变量t=（t_1，t_2，.）在实紧群的积分和酉群的两组时间（t，s）中。关键是这些新的含时积分满足可积族：（i）O（n）和Sp（n）对应于标准Toda格。(ii)U（n）对应于Toeplitz格，离散sinh-Gordon方程的一个非常特殊的简化。两个系统，标准Toda格和Toeplitz格也是2-Toda格的约化，从而导致自然顶点算子，因此，自然Virasoro代数，其子代数零化tau-函数。将这些方程结合起来，就得到了Painlevé V方程。
英文摘要：
Matrix Fourier-like integrals over the classical groups O_+(n), O_-(n), Sp(n) and U(n) are connected with the distribution of the length of the longest increasing sequence in random permutations and random involutions and the spectrum of random matrices. One of the purposes of this paper is to show that all those integrals satisfy the Painlevé V equation with specific initial conditions. In this work, we present both, new results and known ones, in a unified way. Our method consists of inserting one set of time variables t=(t_1,t_2,...) in the integrals for the real compact groups and two sets of times (t,s) for the unitary group. The point is that these new time-dependent integrals satisfy integrable hierarchies: (i) O(n) and Sp(n) correspond to the standard Toda lattice. (ii) U(n) corresponds to the Toeplitz lattice, a very special reduction of the discrete sinh-Gordon equation. Both systems, the standard Toda lattice and the Toeplitz lattice are also reductions of the 2-Toda lattice, thus leading to a natural vertex operator, and so, a natural Virasoro algebra, a subalgebra of which annihilates the tau-functions. Combining these equations leads to the Painlevé V equation for the integrals.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
