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 establish (paying particular attention to large or small ratios $p/n$) that, with careful choices of the rescaling constants and of 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 expansion terms as linear combinations of higher-order derivatives of the limit law $F_\beta$ with rational polynomial coefficients. The parametrizations are fine-tuned so that the expansion coefficients in the Gaussian cases are, for given $n$, the limits $p\to\infty$ of those 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, focusing 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.