Asymptotic Expansions of the Limit Laws of Gaussian and Laguerre (Wishart) Ensembles at the Soft Edge

The large-matrix limit laws of the rescaled largest eigenvalue of the orthogonal, unitary and symplectic $n$-dimensional Gaussian ensembles -- and of the corresponding Laguerre ensembles (Wishart distributions) for various regimes of the parameter $\alpha$ (degrees of freedom $p$) -- are known to be the Tracy-Widom distributions $F_\beta$ ($\beta=1,2,4$). We will establish (paying particular attention to large, or small, ratios $p/n$) that, with careful choices of the rescaling constants and the expansion parameter $h$, the limit laws embed into asymptotic expansions in powers of $h$, where $h \asymp n^{-2/3}$ resp. $h \asymp (n\,\wedge\,p)^{-2/3}$. We find explicit analytic expressions of the first few expansions terms as linear combinations, with rational polynomial coefficients, of higher order derivatives of the limit law $F_\beta$. With a proper parametrization, the expansions in the Gaussian cases can be understood, for given $n$, as the limit $p\to\infty$ of the Laguerre cases. Whereas the results for $\beta=2$ are presented with proof, the discussion of the cases $\beta=1,4$ is based on some hypotheses, focussing on the algebraic aspects of actually computing the polynomial coefficients. For the purposes of illustration and validation, the various results are checked against simulation data with large sample sizes.