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

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.

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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

Information

Received: 1 April 2015; Revised: 1 April 2015; Published: June 2016
First available in Project Euclid: 14 June 2016

zbMATH: 1345.60011
MathSciNet: MR3513610
Digital Object Identifier: 10.1214/15-AAP1121

Subjects:
Primary: 60F05
Secondary: 15B52

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.26 • No. 3 • June 2016
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