2006/09/13 by P. J. Forrester, P J Forrester, N. S. Witte +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Gaussian #Gravitational singularity #Invariant (physics) #Matrix (chemical analysis) #Matrix Theory and Algorithms #Neighbourhood (mathematics) #Nonlinear system #Polynomial and algebraic computation #Random Matrices and Applications #Random matrix #Unitary state #math-ph #math.CA #math.MP #msc:05E35 #msc:33C45 #msc:34M55 #msc:37F10 #msc:39A05
paper · pdf · doi:10.1088/0305-4470/39/39/s14
published as J. Phys. A: Math. Gen. {\bf 39}, 12211-12233 (2006) · Dedicated to the centenary of the publication of the Painlevé VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 1905
openalex publication_date 2006/09/13 · arxiv created 2006/10/16 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realized by matrices with Gaussian entries leads to statistical quantities related to the eigenspectrum, such as the distribution of the largest eigenvalue, which can be expressed as multidimensional integrals or equivalently as determinants. These distributions are well known to be τ-functions for Painlevé systems, allowing for the former to be characterized as the solution of certain nonlinear equations. We consider the random matrix ensembles for which the nonlinear equation is the σ form of P VI . Known results are reviewed, as is their implication by way of series expansions for the distributions. New results are given for the boundary conditions in the neighbourhood of the fixed singularities at t = 0, 1, ∞ of σP VI displayed by a generalization of the generating function for the distributions. The structure of these expansions is related to Jimbo's general expansions for the τ-function of σP VI in the neighbourhood of its fixed singularities, and this theory is itself put in its context of the linear isomonodromy problem relating to P VI .