Communications in Mathematical Physics ( IF 2.2 ) Pub Date : 2021-04-29 , DOI: 10.1007/s00220-021-04059-1 Christophe Charlier 1 , Jonatan Lenells 1 , Julian Mauersberger 1
We consider the limiting process that arises at the hard edge of Muttalib–Borodin ensembles. This point process depends on \(\theta > 0\) and has a kernel built out of Wright’s generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the form
$$\begin{aligned} {\mathbb {P}}(\text{ gap } \text{ on } [0,s]) = C \exp \left( -a s^{2\rho } + b s^{\rho } + c \ln s \right) (1 + o(1)) \qquad \text{ as } s \rightarrow + \infty , \end{aligned}$$where the constants \(\rho \), a, and b have been derived explicitly via a differential identity in s and the analysis of a Riemann–Hilbert problem. Their method can be used to evaluate c (with more efforts), but does not allow for the evaluation of C. In this work, we obtain expressions for the constants c and C by employing a differential identity in \(\theta \). When \(\theta \) is rational, we find that C can be expressed in terms of Barnes’ G-function. We also show that the asymptotic formula can be extended to all orders in s.
中文翻译:
![](https://scdn.x-mol.com/jcss/images/paperTranslation.png)
Muttalib-Borodin 集合在硬边的高阶大间隙渐近
我们考虑在 Muttalib-Borodin 集合的硬边缘出现的限制过程。这个点过程依赖于\(\theta > 0\)并且有一个由 Wright 的广义 Bessel 函数构建的内核。在最近的一篇论文中,Claeys、Girotti 和 Stivigny 已经为这些集合中的大间隙概率建立了一阶和二阶渐近法。这些渐近线的形式为
$$\begin{aligned} {\mathbb {P}}(\text{ gap } \text{ on } [0,s]) = C \exp \left( -as^{2\rho } + bs^{ \rho } + c \ln s \right) (1 + o(1)) \qquad \text{ as } s \rightarrow + \infty , \end{aligned}$$其中常数\(\rho \)、a和b已通过s 中的微分恒等式和黎曼-希尔伯特问题的分析明确导出。他们的方法可用于评估c(需要更多努力),但不允许评估C。在这项工作中,我们通过在\(\theta \) 中使用微分恒等式来获得常数c和C的表达式。当\(\theta \)是有理数时,我们发现C可以用 Barnes' G 表示-功能。我们还表明渐近公式可以扩展到s 中的所有阶数。