A NOTE ON THE EXPANSION OF THE SMALLEST EIGENVALUE DISTRIBUTION OF THE LUE AT THE HARD EDGE

• 成果类型: Article
• 署名作者: Bornemann, Folkmar
• 署名单位: Technical University of Munich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/15-AAP1121
• 发表日期: 2016
• 页码: 1942-1946
• 关键词: gue

摘要

In a recent paper, Edelman, Guionnet and Peche 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.