## The Annals of Applied Probability

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

Folkmar Bornemann

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

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

#### 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

#### References

• [1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Washington, DC.
• [2] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Univ. Press, Cambridge.
• [3] Choup, L. N. (2006). Edgeworth expansion of the largest eigenvalue distribution function of GUE and LUE. Int. Math. Res. Not. Art. ID 61049, 32.
• [4] Edelman, A., Guionnet, A. and Péché, S. (2014). Beyond universality in random matrix theory. Preprint. Available at arXiv:1405.7590.
• [5] Forrester, P. J. (1993). The spectrum edge of random matrix ensembles. Nuclear Phys. B 402 709–728.
• [6] Forrester, P. J. (2010). Log-Gases and Random Matrices. Princeton Univ. Press, Princeton, NJ.
• [7] Johnstone, I. M. and Ma, Z. (2012). Fast approach to the Tracy–Widom law at the edge of GOE and GUE. Ann. Appl. Probab. 22 1962–1988.
• [8] Moecklin, E. (1934). Asymptotische Entwicklungen der Laguerreschen Polynome. Comment. Math. Helv. 7 24–46.
• [9] Schehr, G. (2014). On the smallest eigenvalue at the hard edge of the Laguerre ensemble of complex random matrices: $1/N$ corrections, talk at FOCM’14, December 16, 2014, Montevideo, joint work with Anthony Perret, available at the http://lptms.u-psud.fr/gregory-schehr/files/2014/12/Talk_FOCM.pdf.
• [10] Szegő, G. (1975). Orthogonal Polynomials, 4th ed. Amer. Math. Soc., Providence, RI.
• [11] Tracy, C. A. and Widom, H. (1994). Level spacing distributions and the Bessel kernel. Comm. Math. Phys. 161 289–309.
• [12] Tricomi, F. (1947). Sulle funzioni ipergeometriche confluenti. Ann. Mat. Pura Appl. (4) 26 141–175.