2019/03/21 by Peter J. Forrester, Forrester, Peter J., Allan K. Trinh +1
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1903.08823
openalex publication_date 2019/03/21 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
A fundamental question in random matrix theory is to quantify the optimal rate of convergence to universal laws. We take up this problem for the Laguerre β ensemble, characterised by the Dyson parameter β, and the Laguerre weight xa e-βx/2, x > 0 in the hard edge limit. The latter relates to the eigenvalues in the vicinity of the origin in the scaled variable x ↦ x/4N. Previous work has established the corresponding functional form of various statistical quantities --- for example the distribution of the smallest eigenvalue, provided that a ∈ \mathbb Z≥ 0. We show, using the theory of multidimensional hypergeometric functions based on Jack polynomials, that with the modified hard edge scaling x ↦ x/4(N+a/β), the rate of convergence to the limiting distribution is O(1/N2), which is optimal. In the case β= 2, general a> -1 the explicit functional form of the distribution of the smallest eigenvalue at this order can be computed, as it can for a=1 and general β> 0. An iterative scheme is presented to numerically approximate the functional form for general a ∈ \mathbb Z≥ 2.