Rate of convergence at the hard edge for various P\'olya ensembles of positive definite matrices

TL;DR

This paper analyzes the convergence rates at the hard edge for various Pólya ensembles of positive definite matrices, deriving asymptotic expansions and identifying conditions for faster convergence rates.

Contribution

It provides structured asymptotic expansions for the hard edge kernels of multiple Pólya ensembles, including Laguerre and Jacobi, and identifies conditions for $O(1/N^2)$ convergence.

Findings

Hard edge kernels admit $1/N$ expansions with structured leading terms.

Symmetry of the Bessel kernel enables $O(1/N^2)$ convergence.

Results apply to products of Laguerre ensembles and their inverses.

Abstract

The theory of P\'olya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large $N$ asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib-Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter includes as a special case the Jacobi unitary ensemble. In each case the hard edge scaled kernel permits an expansion in powers of $1/N$, with the leading term given in a structured form involving the hard edge scaling of the biorthogonal pair. The Laguerre and Jacobi ensembles have the special feature that their hard edge scaled kernel -- the Bessel kernel -- is symmetric and this leads to there being a choice of hard edge scaling variables for which the rate of convergence of the correlation functions is $O(1/N^2)$.