The generating function for the Bessel point process and a system of coupled Painleve V equations
The generating function for the Bessel point process and a system of coupled Painleve V equations
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DOI：
10.1142/s2010326319500084
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发表时间：
2019-07-01
影响因子：
0.9
通讯作者：
Doeraene, Antoine
中科院分区：
文献类型：
--
作者：
Charlier, Christophe;Doeraene, Antoine
研究了Bessel点过程中不相交区间上k个占有数的联合概率母函数。这个生成函数可以表示为Fredholm行列式。我们得到一个表达式，它在一个系统的耦合Painleve V方程，这是来自一个Lax对的黎曼-希尔伯特问题。这推广了Tracy和Widom [C. A. Tracy和H. Widom，Level spacing distributions and the Bessel kernel，Commun. Math.Phys.161（2）（1994）289-309]，其对应于k = 1的情况。我们还提供了一些例子和应用。特别地，几个相关的量可以用生成函数来表示，例如不相交的有界区间的并集上的差距概率，两个最小粒子之间的差距，以及具有拉盖尔权重的nxn汉克尔行列式在硬边附近具有几个跳跃间断的大n渐近性。
We study the joint probability generating function for k occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painleve V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. This generalizes a result of Tracy and Widom [C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Commun. Math. Phys. 161(2) (1994) 289-309], which corresponds to the case k = 1. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probability on a union of disjoint bounded intervals, the gap between the two smallest particles, and large n asymptotics for nxn Hankel determinants with a Laguerre weight possessing several jump discontinuities near the hard edge.