Eigenvalues of the classical Gaussian and Laguerre (Wishart) random matrix ensembles, when scaled to be finite in the large $N$ (size) limit, are distributed according to the Wigner semicircle and Marchenko-Pastur laws, respectively. Zooming into the edges of these spectra, one observes that the largest eigenvalues of the Gaussian and Laguerre unitary ensembles follow the Tracy-Widom distribution $Ai'(x)^2 - x Ai(x)^2$, while the smallest eigenvalue of the Laguerre unitary ensemble follows an analogue given in terms of the Bessel function of the first kind. As these edge distributions are limiting quantities, there is some freedom in how one may "zoom in" to see them. However, it turns out there is an optimal choice of scaling and centring parameters that maximises the rates of convergence. In this talk, we will see how these optimal parameters can be found by probing linear differential equations satisfied by the eigenvalue densities of the classical Gaussian and Laguerre ensembles. Moreover, we will see that under these optimal scalings, the eigenvalue densities admit large $N$ expansions with integrable structures that were unknown until the recent work of Folkmar Bornemann and our own follow up work.

Based on joint work with Peter J. Forrester and Bo-Jian Shen.