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

Abstract

The $\beta$ 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 $\beta$ 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 numerical demonstration of this effect.

17 pages, 9 figures

References in corpus (5)

• Growing interfaces uncover universal fluctuations behind scale invariance
• Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy--Widom limits and rates of convergence
• Accuracy of the Tracy--Widom limits for the extreme eigenvalues in white Wishart matrices
• On the spacing distribution of the Riemann zeros: corrections to the asymptotic result
• Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations

Cited by in corpus (1)

• Consecutive level spacings in the chiral Gaussian unitary ensemble: From the hard and soft edge to the bulk