Finite size corrections at the hard edge for the Laguerre $\beta$ ensemble

TL;DR

This paper investigates the rate at which the eigenvalue distribution of the Laguerre beta ensemble converges to its universal limit at the hard edge, establishing an optimal convergence rate of O(1/N^2) using hypergeometric functions and Jack polynomials.

Contribution

It introduces a modified scaling for the Laguerre beta ensemble that achieves the optimal convergence rate and provides explicit formulas and numerical methods for the distribution of the smallest eigenvalue.

Findings

Convergence rate to the limiting distribution is O(1/N^2) under the modified scaling.

Explicit distribution formulas are derived for beta=2 and certain a values.

Numerical approximation scheme for general a is developed.

Abstract

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 $\beta$ ensemble, characterised by the Dyson parameter $\beta$, and the Laguerre weight $x^a e^{-\beta x/2}$, $x > 0$ in the hard edge limit. The latter relates to the eigenvalues in the vicinity of the origin in the scaled variable $x \mapsto x/4N$. Previous work has established the corresponding functional form of various statistical quantities --- for example the distribution of the smallest eigenvalue, provided that $a \in \mathbb Z_{\ge 0}$. We show, using the theory of multidimensional hypergeometric functions based on Jack polynomials, that with the modified hard edge scaling $x \mapsto x/4(N+a/\beta)$, the rate of convergence to the limiting distribution is $O(1/N^2)$, which is optimal. In the case $\beta = 2$, 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 $\beta > 0$. An iterative scheme is presented to numerically approximate the functional form for general $a \in \mathbb Z_{\ge 2}$.