Journal article

Optimal soft edge scaling variables for the Gaussian and Laguerre even beta ensembles

Peter J Forrester, Allan K Trinh



The β ensembles are a class of eigenvalue probability densities which generalise the invariant ensembles of classical random matrix theory. In the case of the Gaussian and Laguerre weights, the corresponding eigenvalue densities are known in terms of certain β dimensional integrals. We study the large N asymptotics of the density with a soft edge scaling. In the Laguerre case, this is done with both the parameter a fixed, and with a proportional to N. It is found in all these cases that by appropriately centring the scaled variable, the leading correction term to the limiting density is O(N−2/3). A known differential-difference recurrence from the theory of Selberg integrals allows for a num..

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University of Melbourne Researchers


Awarded by ARC grant

Funding Acknowledgements

This work is part of a research program supported by the Australian Research Council (ARC) through the ARC Centre of Excellence for Mathematical and Statistical frontiers (ACEMS). PJF also acknowledges partial support from ARC grant DP170102028, and AKT acknowledges the support of a Melbourne postgraduate award.