Muttalib-Borodin系综的向量Riemann-Hilbert方法
A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles
摘要：
本文考虑Laguerre型的Muttalib-Borodin系综，它是$[0,\infty)$上的一个决定点过程，它依赖于变权$x^αe^{-nV(x)}$，$α>-1$和一个参数$θ$ .对于$θ$是一个正整数，我们得到了相关的双正交多项式的渐近附近的原点为$n\to \infty$的一大类势函数$V$。这进一步使我们能够建立相关核的硬边缘缩放极限，这是以前只知道在特殊情况下，并证明是普遍的。我们的证明是基于Deift/Zhou非线性最速下降分析的两个$1 \times 2$向量值黎曼-希尔伯特问题的特征的双正交多项式和显式结构的$(θ+1)\times(θ+1)$局部参数在原点附近的Meijer G-函数。
英文摘要：
In this paper, we consider Muttalib-Borodin ensemble of Laguerre type, a determinantal point process over $[0,\infty)$ which depends on the varying weights $x^αe^{-nV(x)}$, $α>-1$, and a parameter $θ$. For $θ$ being a positive integer, we derive asymptotics of the associated biorthogonal polynomials near the origin for a large class of potential functions $V$ as $n\to \infty$. This further allows us to establish the hard edge scaling limit of the correlation kernel, which is previously only known in the special cases and conjectured to be universal. Our proof is based on the Deift/Zhou nonlinear steepest descent analysis of two $1 \times 2$ vector-valued Riemann-Hilbert problems that characterize the biorthogonal polynomials and the explicit constructions of $(θ+1)\times(θ+1)$ local parametrices near the origin in terms of the Meijer G-functions.
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