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 a sample size of a thousand million.

翻译：正交、酉和辛 $n$ 维高斯系综（以及对应拉盖尔系综（Wishart 分布）在参数 $\alpha$（自由度 $p$）的不同取值区域下）重标最大特征值的大矩阵极限律已知为 Tracy-Widom 分布 $F_\beta$（$\beta=1,2,4$）。我们将在本文中证明（特别关注大或小比值 $p/n$ 的情形）：通过精心选择重标常数和展开参数 $h$，极限律可嵌入 $h$ 幂次的渐近展开中，其中 $h \asymp n^{-2/3}$ 或 $h \asymp (n\,\wedge\,p)^{-2/3}$。我们发现前几项展开项可显式解析表达为极限律 $F_\beta$ 高阶导数的线性组合，其系数为有理多项式。在适当参数化下，高斯情形的展开对给定 $n$ 可理解为拉盖尔情形在 $p\to\infty$ 时的极限。$\beta=2$ 的结果附有证明，而 $\beta=1,4$ 的讨论基于若干假设，重点在于多项式系数的代数计算。为便于说明与验证，各项结果与十亿样本量的模拟数据进行了对比检验。