We continue our work [arXiv:2403.07628] on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre random matrix ensembles. By revisiting the construction of the associated skew-orthogonal polynomials in terms of wave functions, we obtain concise expressions for the level densities that are well suited for proving asymptotic expansions in powers of a certain parameter $h \asymp n^{-2/3}$. In the unitary case, the expansion for the level density can be used to reconstruct the first correction term in an established asymptotic expansion of the associated generating function. In the orthogonal and symplectic cases, we can even reconstruct the conjectured first and second correction terms.

翻译：我们延续先前工作[arXiv:2403.07628]中关于经典$n$维高斯与拉盖尔随机矩阵系综在软边界的渐近展开研究。通过基于波函数重新审视相关斜交多项式的构造，我们得到了适用于证明以参数$h \asymp n^{-2/3}$的幂次形式进行渐近展开的能级密度简明表达式。在酉系综情形中，能级密度的展开可用于重构相关生成函数已知渐近展开中的首项修正项。在正交与辛系综情形中，我们甚至能够重构理论预测的首项与次项修正项。