Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices

Abstract

The theory of Pólya 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 $O(1/N^2)$.

Keywords:

• Convergence rate
• Pólya ensemble
• special function

2010 Mathematics Subject Classifications:

• 15B52
• 15A15
• 33E20

Acknowledgments

We thank Mario Kieburg for feedback on a draft of this work and anonymous referees for helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is part of the program of study supported by the Australian Research Council (ARC) Centre of Excellence ACEMS. The work of P.J.F. was also partially supported by the ARC grant DP210102887.