Perturbed Hankel determinant, correlation functions and Painlevé equations

Abstract

In this paper, we continue with the study of the Hankel determinant, generated by a Pollaczek-Jacobi type weight, w(x; t, α, β) ≔ xα(1 − x)βe−t/x, x ∈ [0, 1], α > 0, β > 0, t ≥ 0. This reduces to the “pure” Jacobi weight at t = 0. It was shown in the work of Chen and Dai [J. Approximation Theory 162(2), 2149–2167 (2010)] that the logarithmic derivative of this Hankel determinant satisfies a Jimbo-Miwa-Okamoto σ-form of Painlevé V (PV). We show that, under a double scaling, where n the dimension of the Hankel matrix tends to ∞ and t tends to 0, such that s ≔ 2n2t is finite, the double scaled Hankel determinant (effectively an operator determinant) has an integral representation in terms of a particular PIII′. Expansions of the scaled Hankel determinant for small and large s are found. We also consider another double scaling with α = − 2n + λ, where n → ∞, and t tends to 0, such that s ≔ nt is finite. In this situation, the scaled Hankel determinant has an integral representation in terms of a particular PV, and its small and large s asymptotic expansions are also found. The reproducing kernel in terms of monic polynomials orthogonal with respect to the Pollaczek-Jacobi type weight under the origin (or hard edge) scaling may be expressed in terms of the solutions of a second order linear ordinary differential equation (ODE). With special choices of the parameters, the limiting (double scaled) kernel and the second order ODE degenerate to Bessel kernel and the Bessel differential equation, respectively.