Abstract
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 [12] 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 [22], depth [7, 24], and graph distance [9, 15] of 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.
Funding Statement
Bas Lodewijks has been supported by grant GrHyDy ANR-20-CE40-0002.
Acknowledgments
Bas Lodewijks would like to thank Laura Eslava for some useful discussions related to the Kingman n-coalescent and for providing the source code of the figures in this paper.
He would also like to thank the anonymous referees for providing helpful suggestions which led to an improved presentation of the results and proofs.
Citation
Bas Lodewijks. "On joint properties of vertices with a given degree or label in the random recursive tree." Electron. J. Probab. 27 1 - 45, 2022. https://doi.org/10.1214/22-EJP877
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