The Annals of Applied Probability

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

Folkmar Bornemann

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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.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 3 (2016), 1942-1946.

Dates
Received: April 2015
Revised: April 2015
First available in Project Euclid: 14 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1465905023

Digital Object Identifier
doi:10.1214/15-AAP1121

Mathematical Reviews number (MathSciNet)
MR3513610

Zentralblatt MATH identifier
1345.60011

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 15B52: Random matrices

Keywords
Rate of convergence random matrix smallest eigenvalue LUE

Citation

Bornemann, Folkmar. A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge. Ann. Appl. Probab. 26 (2016), no. 3, 1942--1946. doi:10.1214/15-AAP1121. https://projecteuclid.org/euclid.aoap/1465905023


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References

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