The theory of Pólya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large N asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib–Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter includes, as a special case, the Jacobi unitary ensemble. In each case, the hard edge scaled kernel permits an expansion in powers of 1/N, with the leading term given in a structured form involving the hard-edge scaling of the biorthogonal pair. The Laguerre and Jacobi ensembles have the special feature that their hard edge scaled kernel – the Bessel kernel – is symmetric and this leads to there being a choice of hard edge scaling variables for which the rate of convergence of the correlation functions is $O(1/N^2)$.

正定随机矩阵的 Pólya 系综理论提供了相应的双正交对和相关核的结构公式，非常适合计算硬边大N渐近线。对 Laguerre 系综的产物、Laguerre Muttalib-Borodin 系综以及 Laguerre 系综及其逆系的产物进行了这种分析。作为特例，后者包括 Jacobi 酉系综。在每种情况下，硬边缘缩放内核允许以 1/ N的幂次扩展, 前导项以结构化形式给出，涉及双正交对的硬边缩放。Laguerre 和 Jacobi 系综的特点是它们的硬边缘缩放核——贝塞尔核——是对称的，这导致可以选择相关函数的收敛速度为的硬边缘缩放变量$○(1/{ñ}^2)$.