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

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.