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In a recent paper, Edelman, Guionnet and P\'{e}ch\'{e} conjectured aparticular $n^{-1}$ correction term of the smallest eigenvalue distribution ofthe Laguerre unitary ensemble (LUE) of order $n$ in the hard-edge scalinglimit: specifically, the derivative of the limit distribution, that is, thedensity, shows up in that correction term. We give a short proof by modifyingthe hard-edge scaling to achieve an optimal $O(n^{-2})$ rate of convergence ofthe smallest eigenvalue distribution. The appearance of the derivative followsthen by a Taylor expansion of the less optimal, standard hard-edge scaling. Werelate the $n^{-1}$ correction term further to the logarithmic derivative ofthe Bessel kernel Fredholm determinant in the work of Tracy and Widom.

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