A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge

In a recent paper, Edelman, Guionnet and Péché conjectured a particular n⁻¹ correction term of the smallest eigenvalue distribution of the Laguerre unitary ensemble (LUE) of order n in the hard-edge scaling limit: specifically, the derivative of the limit distribution, that is, the density, shows up in that correction term. We give a short proof by modifying the hard-edge scaling to achieve an optimal O(n⁻²) rate of convergence of the smallest eigenvalue distribution. The appearance of the derivative follows then by a Taylor expansion of the less optimal, standard hard-edge scaling. We relate the n⁻¹ correction term further to the logarithmic derivative of the Bessel kernel Fredholm determinant in the work of Tracy and Widom.