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 $i$th 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).