Abstract
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.
Citation
Folkmar Bornemann. "A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge." Ann. Appl. Probab. 26 (3) 1942 - 1946, June 2016. https://doi.org/10.1214/15-AAP1121
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