N. S. Witte

Application of the τ -Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE

Abstract: Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of , where for and otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of . Of particular interest are and FN(λ;2), and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto τ-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities and FN(λ;a) for the other classical matrix ensembles.

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

Okamoto has obtained a sequence of

Gap probabilities in the finite and scaled Cauchy random matrix ensembles

\tau

Finite one-dimensional impenetrable Bose systems: Occupation numbers

-functions for the \PVI 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 multi-dimensional 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

Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

N

Discrete Painlevé equations and random matrix averages

, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the \PVI theory. 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

Random matrix theory and the sixth Painlevé equation

a

τ-function evaluation of gap probabilities in orthogonal and symplectic matrix ensembles

a non-negative integer) and Laguerre symplectic ensemble (LSE) (parameter

Exact Wigner Surmise Type Evaluation of the Spacing Distribution in the Bulk of the Scaled Random Matrix Ensembles

a

Painleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons

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