Painleve ́ III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight

In this paper, we consider the Hankel determinants associated with the singu-larly perturbed Laguerre weight w(x) = xαe−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 Painleve ́ III transcendent is involved in the approximation, as well as in the large-n asymp-totics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic re-sults in an earlier paper of the authors, obtained by using the Deift-Zhou nonlinear steepest descent method