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

作者：Bornemann Folkmar^*

来源：Annals of Applied Probability, 2016, 26(3): 1942-1946.

DOI：10.1214/15-AAP1121

• Rate of convergence
• random matrix
• smallest eigenvalue
• LUE

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.

• 出版日期2016-6