标题： 关于具有不连续性的合流超几何核的弗雷德霍姆行列式 显示英文标题

标题： On the Fredholm determinant of the confluent hypergeometric kernel with discontinuities

摘要： 我们考虑具有合流超几何核的行列式点过程。 该过程是随机矩阵理论中的普遍点过程，描述了大型随机厄米特矩阵在Fisher-Hartwig奇异性附近的特征值分布。 应用黎曼-希尔伯特方法，我们研究了该过程在任意给定区间数量上的生成函数。 它可以表示为具有$n$个不连续性的合流超几何核的弗雷德霍姆行列式。 在本文中，我们通过使用耦合的皮亚诺V系统的哈密顿量，推导出该行列式的积分表示。 通过计算哈密顿量的总积分，我们得到了当$n$个不连续性趋于无穷大时行列式的渐近行为，包括常数项。 这里的常数项用巴内斯$G$函数来表示。 显示英文摘要

摘要： We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we study the generating function of this process on any given number of intervals. It can be expressed as the Fredholm determinant of the confluent hypergeometric kernel with $n$ discontinuities. In this paper, we derive an integral representation for the determinant by using the Hamiltonian of the coupled Painlev\'e V system. By evaluating the total integral of the Hamiltonian, we obtain the asymptotics of the determinant as the $n$ discontinuities tend to infinity up to and including the constant term. Here the constant term is expressed in terms of the Barnes $G$-function.