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    Application of the tau-function theory of Painleve equations to random matrices: P-VI, the JUE, CyUE, cJUE and scaled limits

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    Author
    Forrester, PJ; Witte, NS
    Date
    2004-06-01
    Source Title
    NAGOYA MATHEMATICAL JOURNAL
    Publisher
    CAMBRIDGE UNIV PRESS
    University of Melbourne Author/s
    Forrester, Peter; WITTE, NICHOLAS
    Affiliation
    Mathematics And Statistics
    Metadata
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    Document Type
    Journal Article
    Citations
    Forrester, P. J. & Witte, N. S. (2004). Application of the tau-function theory of Painleve equations to random matrices: P-VI, the JUE, CyUE, cJUE and scaled limits. NAGOYA MATHEMATICAL JOURNAL, 174, pp.29-114. https://doi.org/10.1017/S0027763000008801.
    Access Status
    This item is currently not available from this repository
    URI
    http://hdl.handle.net/11343/26127
    DOI
    10.1017/S0027763000008801
    Description

    C1 - Journal Articles Refereed

    Abstract
    <jats:title>Abstract</jats:title><jats:p>Okamoto has obtained a sequence of <jats:italic>τ</jats:italic>-functions for the P<jats:sub>VI</jats:sub> system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be re-expressed as multidimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter <jats:italic>N</jats:italic>, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the P<jats:sub>VI</jats:sub> theory. We show that the Hamiltonian also satisfies an equation related to the discrete P<jats:sub>V</jats:sub> equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general P<jats:sub>V</jats:sub> transcendent in <jats:italic>σ</jats:italic> form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter <jats:italic>a</jats:italic> a non-negative integer) and Laguerre symplectic ensemble (LSE) (parameter <jats:italic>a</jats:italic> an even non-negative integer) as finite dimensional combinatorial integrals over the symplectic and orthogonal groups respectively; to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to the <jats:italic>τ</jats:italic>-function evaluation of the largest eigenvalue in the finite LOE and LSE with parameter <jats:italic>a =</jats:italic> 0; and to the characterisation of the diagonal-diagonal spin-spin correlation in the two-dimensional Ising model.</jats:p>
    Keywords
    Differential; Difference and Integral Equations; Dynamical Systems; Probability Theory ; Mathematical Sciences; Physical Sciences

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