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Finite‐size corrections at the hard edge for the Laguerre β ensemble
Author(s) -
Forrester Peter J.,
Trinh Allan K.
Publication year - 2019
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12279
Subject(s) - mathematics , laguerre polynomials , eigenvalues and eigenvectors , hypergeometric function , rate of convergence , mathematical analysis , random matrix , hermite polynomials , convergence (economics) , laguerre's method , orthogonal polynomials , statistical physics , quantum mechanics , physics , channel (broadcasting) , economic growth , electrical engineering , economics , engineering , classical orthogonal polynomials
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, characterized by the Dyson parameter β, and the Laguerre weightx a e − β x / 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 / 4 N . Previous work has established the corresponding functional form of various statistical quantities—for example, the distribution of the smallest eigenvalue, provided that a ∈ 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 / N 2 ) , 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 ∈ Z ≥ 2.