1999/12/17 by Mark Adler, Pierre van Moerbeke, Adler, Mark +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #math-ph #math.CO #math.MP #math.PR #nlin.SI
paper · pdf · doi:10.48550/arxiv.math/9912143
57 pages
openalex publication_date 1999/12/17 · arxiv created 2000/09/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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=(t1,t2,...) 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.