2002/03/25 by Peter J. Forrester, P. J. Forrester, N. S. Witte · 25 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Applied mathematics #Circular ensemble #Differential equation #Eigenvalues and eigenvectors #Function (biology) #Generating function #Geometry #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Matrix function #Product (mathematics) #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Symmetric matrix #Symmetry (geometry) #Symplectic geometry #Symplectic manifold #Symplectic matrix #Unitary group #Unitary matrix #Unitary state #math-ph #math.MP #msc:15A52 #msc:33E17 #msc:58F07
paper · pdf · doi:10.1088/0951-7715/15/3/325
published in Nonlinearity 15(3), 937-954 (IOP Publishing) · AMS-Latex
arxiv created 2002/03/25 · openalex publication_date 2002/04/18 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact τ-functions for certain Painlevé systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise τ-functions for certain Painlevé systems. In the case of symplectic matrix ensembles, all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two τ-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two τ-functions gives the gap probability in the corresponding unitary symmetry case, while one of these τ-functions is the gap probability in the corresponding orthogonal symmetry case.