Convergence and compactness of discrete aggregation trees
Abstract
We study a class of random aggregation trees that generalizes the discrete stick-breaking construction of the uniform labelled tree (Aldous, 1991). To sample the tree of size n, vertices are added sequentially, with the ith vertex starting a new branch with a prescribed probability f(n,i); otherwise, it extends the current branch. We also define a continuum analogue by replacing the Poisson point process of intensity tdt in the construction of the Brownian continuum random tree by one of intensity f(t)dt. We establish scaling limits for two families of discrete aggregation trees in the Gromov-Hausdorff-Prokhorov topology. When f(n,i)=(i/n)β, with β>0, after rescaling the graph distance by n−β/(β+1), the random tree converges to the compact aggregation tree with f(t)=tβ. This recovers convergence to the Brownian continuum random tree when β=1 (Aldous, 1991), as well as scaling limits of choice spanning trees for integer β (Archer and Shalev, 2024). We also prove convergence under rescaling to the compact aggregation tree with f(t)=logγ(1+t) for every γ>1. Finally, we identify necessary conditions for compactness. Consequently, the threshold γ>1 in the logarithmic family is sharp. These results provide insight into an open problem on compactness criteria for random aggregation trees (Curien and Haas, 2014).
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BibTeX
@misc{donderwinkel2026convergence,
title = {{Convergence and compactness of discrete aggregation trees}},
author = {Serte Donderwinkel and Jeroen van Haastert},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13066},
eprint = {2609.13066},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13066},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Convergence and compactness of discrete aggregation trees
AU - Serte Donderwinkel
AU - Jeroen van Haastert
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13066
UR - https://arxiv.org/abs/2609.13066
AB - We study a class of random aggregation trees that generalizes the discrete stick-breaking construction of the uniform labelled tree (Aldous, 1991). To sample the tree of size n, vertices are added sequentially, with the ith vertex starting a new branch with a prescribed probability f(n,i); otherwise, it extends the current branch. We also define a continuum analogue by replacing the Poisson point process of intensity tdt in the construction of the Brownian continuum random tree by one of intensity f(t)dt. We establish scaling limits for two families of discrete aggregation trees in the Gromov-Hausdorff-Prokhorov topology. When f(n,i)=(i/n)^β, with β>0, after rescaling the graph distance by n^−β/(β+1), the random tree converges to the compact aggregation tree with f(t)=t^β. This recovers convergence to the Brownian continuum random tree when β=1 (Aldous, 1991), as well as scaling limits of choice spanning trees for integer β (Archer and Shalev, 2024). We also prove convergence under rescaling to the compact aggregation tree with f(t)=log^γ(1+t) for every γ>1. Finally, we identify necessary conditions for compactness. Consequently, the threshold γ>1 in the logarithmic family is sharp. These results provide insight into an open problem on compactness criteria for random aggregation trees (Curien and Haas, 2014).
ER -