Finite‐size corrections at the hard edge for the Laguerre β ensemble
| Autor: | Peter J. Forrester, Allan K. Trinh |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Applied Mathematics
010102 general mathematics Asymptotic distribution 01 natural sciences 010104 statistics & probability Distribution (mathematics) Rate of convergence Laguerre polynomials Applied mathematics 0101 mathematics Hypergeometric function Random matrix Eigenvalues and eigenvectors Mathematics Variable (mathematics) |
| Zdroj: | Studies in Applied Mathematics. 143:315-336 |
| ISSN: | 1467-9590 0022-2526 |
| DOI: | 10.1111/sapm.12279 |
| Popis: | 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 weight (Formula presented.), (Formula presented.) in the hard edge limit. The latter relates to the eigenvalues in the vicinity of the origin in the scaled variable (Formula presented.). Previous work has established the corresponding functional form of various statistical quantities—for example, the distribution of the smallest eigenvalue, provided that (Formula presented.). We show, using the theory of multidimensional hypergeometric functions based on Jack polynomials, that with the modified hard edge scaling (Formula presented.), the rate of convergence to the limiting distribution is (Formula presented.), which is optimal. In the case (Formula presented.), general (Formula presented.) the explicit functional form of the distribution of the smallest eigenvalue at this order can be computed, as it can for (Formula presented.) and general (Formula presented.). An iterative scheme is presented to numerically approximate the functional form for general (Formula presented.). |
| Databáze: | OpenAIRE |
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