Application of the τ‐function theory of Painlevé equations to random matrices: PV, PIII, the LUE, JUE, and CUE

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Abstract
With 〈·〉 denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is for I = (0, s) and I = (s, ∞), where χ = 1 for λl ∈ I and χ = 0 otherwise. Using Okamoto's development of the theory of the Painlevé V equation, it is shown that ẼN(I; a, μ) is a τ‐function associated with the Hamiltonian therein, and so can be characterized as the solution of a certain second‐order second‐degree differential equation, or in terms of the solution of certain difference equations. The cases μ = 0 and μ = 2 are of particular interest, because they correspond to the cumulative distribution and density function, respectively, for the smallest and largest eigenvalue. In the case I = (s, ∞), ẼN(I; a, μ) is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group, and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard‐edge and soft‐edge scaled limits of ẼN(I; a, μ). In particular, in the hard‐edge scaled limit it is shown that the limiting quantity Ẽhard((O, s); a, μ) can be evaluated as a τ‐function associated with the Hamiltonian in Okamoto's theory of the Painlevé III equation.

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