No abstract

In this paper, we study the singularly perturbed Laguerre unitary ensemblewith V t (x) = x + t/x, x ∈ (0, +∞) and t > 0. Due to the effect of t/x for varying t, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves ψ-functions associated with a special solution to a new thirdorder nonlinear differential equation, which is then shown equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter t changes in a finite interval (0, d]. Our approach is based on Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems.

We study the uniform asymptotics of the polynomials orthogonal with respect to analytic weights with jump discontinuities on the real axis, and the influence of the discontinuities on the asymptotic behavior of the recurrence coefficients. The Riemann–Hilbert approach, also termed the Deift–Zhou steepest descent method, is used to derive the asymptotic results. We take as an example the perturbed Gaussian weight , where θ(x) takes the value of 1 for x < 0, and a nonnegative complex constant ω elsewhere, and  as . That is, the jump occurs at the edge of the support of the equilibrium measure. The derivation is carried out in the sense of a double scaling limit, namely,  and . A crucial local parametrix at the edge point where the jump occurs is constructed out of a special solution of the Painlevé XXXIV equation. As a main result, we prove asymptotic formulas of the recurrence coefficients in terms of a special Painlevé XXXIV transcendent under the double scaling limit. The special thirty‐fourth Painlevé transcendent is shown free of poles on the real axis. A consistency check is made with the reduced case when ω= 1, namely the Gaussian weight: the polynomials in this case are the classical Hermite polynomials. A comparison is also made of the asymptotic results for the recurrence coefficients between the case when the jump happens at the edge and the case with jump inside the support of the equilibrium measure. The comparison provides a formal asymptotic approximation of the Painlevé XXXIV transcendent at positive infinity.

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight w(x) = x α e −x−t/x , x ∈ (0, ∞), t > 0 and α > 0. When the matrix size n → ∞, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for t ∈ (0, d], d > 0 fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large-n asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift-Zhou nonlinear steepest descent method.