We conclude our work [arXiv:2403.07628, arXiv:2503.12644] on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre ensembles, now studying the gap-probability generating functions. We show that the correction terms in the asymptotic expansion are multilinear forms of the higher-order derivatives of the leading-order term, with certain rational polynomial coefficients that are independent of the dummy variable. In this way, the same multilinear structure, with the same polynomial coefficients, is inherited by the asymptotic expansion of any linearly induced quantity such as the distribution of the $k$-th largest level. Whereas the results for the unitary ensembles are presented with proof, the discussion of the orthogonal and symplectic ones is based on some hypotheses. To substantiate the hypotheses, we check the result for the $k$-th largest level in the orthogonal ensembles against simulation data for choices of $n$ and $k$ that require as many as four correction terms to achieve satisfactory accuracy.

翻译：本文完成了关于经典$n$维高斯与拉盖尔系综在软边界处渐近展开的研究工作[arXiv:2403.07628, arXiv:2503.12644]，重点探讨了间隙概率生成函数。我们证明了渐近展开中的修正项是主导项高阶导数的多重线性形式，其系数为与哑变量无关的有理多项式。因此，任何线性导出量（例如第$k$大能级的分布）的渐近展开均继承相同的多重线性结构与多项式系数。酉系综的结果已给出证明，而正交与辛系综的讨论基于若干假设。为验证假设，我们将正交系综中第$k$大能级的结果与模拟数据进行了对比，所选$n$和$k$值最多需要四项修正项才能达到满意的精度。