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.

翻译：正交、酉及辛$n$维高斯系综重标度后最大特征值的大矩阵极限律——以及对应参数$\alpha$（自由度$p$）在不同取值区间的拉盖尔系综（威沙特分布）——已知为Tracy-Widom分布$F_\beta$（$\beta=1,2,4$）。本文将证明（特别关注较大或较小的比值$p/n$）：通过精心选择重标度常数与展开参数$h$（其中$h \asymp n^{-2/3}$或$h \asymp (n\,\wedge\,p)^{-2/3}$），这些极限律可嵌入$h$幂次的渐近展开式。我们给出了前几项展开项的显式解析表达式，它们是以极限律$F_\beta$的高阶导数为基、具有有理多项式系数的线性组合。通过恰当参数化，高斯情形的展开式可在给定$n$时理解为拉盖尔情形$p\to\infty$的极限。虽然$\beta=2$的结果已附证明给出，但对$\beta=1,4$情形的讨论基于若干假设，着重探讨多项式系数实际计算的代数问题。为说明与验证结果，本文通过大样本模拟数据对所有结论进行了检验。