论文标题
在随机递归树中具有给定程度或标签的顶点的联合性质上
On joint properties of vertices with a given degree or label in the random recursive tree
论文作者
论文摘要
在本文中，我们研究了随机递归树中高度顶点之间的程度，深度和图形和图形距离的联合行为。我们概括了通过Eslava获得的结果，并将其扩展到包括高度顶点之间的标签和图形距离。对这两种高度顶点的两种特性的分析都是新颖的，特别是与此类顶点深度的行为有关。顺便说一句，我们还获得了带有规定标签的任何固定数量的顶点之间的程度和深度和图形距离的联合行为的结果。这结合了关于顶点之间的程度和深度和图形距离的几个孤立结果，以及文献中已经存在的规定标签。此外，我们将这些结果扩展为共同保留任何数量的固定顶点，并通过提供对分布限制的更详细的描述来改善这些结果。我们的分析基于随机递归树与金曼$ n $ coalescent的表示之间的对应关系。
In this paper, we study the joint behaviour of the degree, depth and label of and graph distance between high-degree vertices in the random recursive tree. We generalise the results obtained by Eslava and extend these to include the labels of and graph distance between high-degree vertices. The analysis of both these two properties of high-degree vertices is novel, in particular in relation to the behaviour of the depth of such vertices. In passing, we also obtain results for the joint behaviour of the degree and depth of and graph distance between any fixed number of vertices with a prescribed label. This combines several isolated results on the degree and depth of and graph distance between vertices with a prescribed label already present in the literature. Furthermore, we extend these results to hold jointly for any number of fixed vertices and improve these results by providing more detailed descriptions of the distributional limits. Our analysis is based on a correspondence between the random recursive tree and a representation of the Kingman $n$-coalescent.
