Random Unitary Matrices, Permutations and Painleve

Integrals over classical Groups, Random permutations, Toda and Toeplitz lattices

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…

Symmetrized random permutations

Selecting N random points in a unit square corresponds to selecting a random permutation. By putting 5 types of symmetry restrictions on the points, we obtain subsets of permutations : involutions, signed permutations and signed involutions. We are interested in the statistics of the length of the longest up/right path of random points selections in each symmetry type as the number of points increases to infinity. The limiting distribution functions are expressed in terms of Painlevé II equation. Some of them are Tracy-Widom distributions in random matrix theory, while there are two new classe…

Integrals over Grassmannians and Random permutations

In testing the independence of two Gaussian populations, one computes the distribution of the sample canonical correlation coefficients, given that the actual correlation is zero. The "Laplace transform" of this distribution is not only an integral over the Grassmannian of p-dimensional planes in complex n-space, but is also related to a generalized hypergeometric function. Such integrals are solutions of Painlevé-like equations. They also have expansions, related to random words of length l formed with an alphabet of p letters. Given that each letter appears in the word, the maximal length of…

The covering radii of the $2$-transitive unitary, Suzuki, and Ree groups

We study the covering radii of $2$-transitive permutation groups of Lie rank one, giving bounds and links to finite geometry.

高斯随机矩阵的渐近自由独立性与元素置换：第二部分：无穷小自由

Asymptotic free independence and entry permutations for Gaussian random matrices. Part II: Infinitesimal freeness

本文研究了具有置换元素的GUE矩阵的渐近无穷小分布，证明了随机均匀置换下GUE矩阵的零无穷小分布及不同置换间的无穷小自由性，并指出GUE矩阵与其转置非无穷小自由。

Permutations without long decreasing subsequences and random matrices

We study the shape of the Young diagram λassociated via the Robinson-Schensted-Knuth algorithm to a random permutation in S_n such that the length of the longest decreasing subsequence is not bigger than a fixed number d; in other words we study the restriction of the Plancherel measure to Young diagrams with at most d rows. We prove that in the limit n\to\infty the rows of λbehave like the eigenvalues of a certain random matrix (traceless Gaussian Unitary Ensemble) with d rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asy…