标题： 高斯和拉盖尔系综在软边的渐近展开 III：生成函数 显示英文标题

标题： Asymptotic Expansions of Gaussian and Laguerre Ensembles at the Soft Edge III: Generating Functions

摘要： 我们总结了我们在[arXiv:2403.07628, arXiv:2503.12644]中关于经典$n$维高斯和拉盖尔系综在软边的渐近展开的工作，现在研究间隙概率生成函数。我们证明渐近展开中的修正项是主导项的高阶导数的多重线性形式，具有某些与哑变量无关的有理多项式系数。以这种方式，任何线性诱导量（如$k$大的水平分布）的渐近展开都继承了相同的多重线性结构和相同的多项式系数。虽然单位系综的结果给出了证明，但正交和辛系综的讨论是基于一些假设。为了验证这些假设，我们将正交系综中$k$大的水平结果与需要多达四个修正项才能达到满意精度的$n$和$k$的选择进行模拟数据对比。 显示英文摘要

摘要： 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.