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

Abstract

With · denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study isfor I = (0, s) and I = (s, ∞), where χ (l) I = 1 for λ l ∈ I and χ (l) I = 0 otherwise. Using Okamoto's development of the theory of the Painlevé V equation, it is shown thatẼ N (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, ∞),Ẽ N (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 softedge scaled limits ofẼ N (I ; a, µ). In particular, in the hard-edge scaled limit it is shown that the limiting quantityẼ hard ((0, s); a, µ) can be evaluated as a τ -function associated with the Hamiltonian in Okamoto's theory of the Painlevé III equation.

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

“…It is known that the simplest correlation functions, i.e. the moments of the characteristic polynomials, can be described in terms of non-linear differential equations (see works of Forrester and White [23,24], and also the paper by Kanzieper [41], and by Splittorff and Verbaarschot [48]). As for more complicated correlation functions a description in terms of differential equations is unknown.…”

Boundary conditions associated with the Painlevé III′ and V evaluations of some random matrix averages

“…The latter is known form our work [3] to be a special case ofẼ hard N (t; a, µ; ξ). The coefficients in the power series appear in an asymptotic formula obtained recently by Conrey, Rubinstein and Snaith [1] for the integer moments of the derivative of the characteristic polynomial of a unitary random matrix.…”

Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as -Functions – Spectrum Singularity Case

“…There is an interesting special case when the τ -functions are classical solutions and this was found to occur in the studies [15], [7] for a ∈ Z ≥0 + 1 2 . For the τ -functions with parameters on the diagonal (a = n − 1 2 , n ∈ Z >0 ) = e X det J j−k (2 √ X) j,k=0,...,n−1 , where J ν (z), I ν (z) are the Bessel function and modified Bessel function respectively.…”