2015/04/30 by Folkmar Bornemann
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Bessel function #Derivative (finance) #Distribution (mathematics) #Eigenvalues and eigenvectors #Kernel (algebra) #Logarithm #Mathematical functions and polynomials #Random Matrices and Applications #Scaling #Taylor series #Term (time) #math-ph #math.MP #math.PR
paper · pdf · doi:10.1214/15-aap1121
published as Annals of Applied Probability 2016, Vol. 26, No. 3, 1942-1946 · Published at http://dx.doi.org/10.1214/15-AAP1121 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2016/06/01 · openalex created_date 2016/06/24 · arxiv created 2016/06/27 · arxiv updated 2016/06/28 · openalex updated_date 2026/08/06
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.