The Muttalib–Borodin ensemble is a probability density function for n particles on the positive real axis that depends on a parameter θ and a weight w. We consider a varying exponential weight that depends on an external field V. In a recent article, the large n behavior of the associated correlation kernel at the hard edge was found for , where only few restrictions are imposed on V. In the current article we generalize the techniques and results of this article to obtain analogous results for , where r is a positive integer. The approach is to relate the ensemble to a type II multiple orthogonal polynomial ensemble with r weights, which can then be related to an (r + 1) (r + 1) Riemann–Hilbert problem. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

Muttalib-Borodin 系综是n 个粒子在正实轴上的概率密度函数，它取决于参数θ和权重w。我们考虑取决于外部场V的变化指数权重。在最近的一篇文章中，发现了相关联的相关内核在硬边缘的大n行为，其中仅对V施加了很少的限制。在当前文章中，我们概括了本文的技术和结果以获得类似的结果，其中r是正整数。该方法是将系综与 II 类多重正交多项式系综相关联r 个权重，然后可以与 ( r + 1) ( r + 1) 黎曼-希尔伯特问题相关。原点周围的局部参数是使用 Meijer G函数构造的。我们将原点周围的局部参数与具有双重匹配的全局参数进行匹配，这是最近引入的一种技术。