Scaling limit of trees with vertices of fixed degrees and heights

We consider large uniform random trees where we fix for each vertex its degree and height. We prove, under natural conditions of convergence for the profile, that those trees properly renormalized converge. To this end, we study the paths from random vertices to the root using coalescent processes. As an application, we obtain scaling limits of Bienaymé-Galton-Watson trees in varying environment.

💡 Research Summary

The paper introduces a novel model of large uniform random trees in which both the degree (number of children) and the height of every vertex are prescribed in advance. For each integer n a plane rooted tree Tₙ is built height‑by‑height: at level i there are Dₙ,i vertices, each vertex (i,j) carries a degree dₙ,i,j, and the children of level i+1 are attached uniformly at random to the vertices of level i that still have free “slots”. The construction guarantees that every vertex is connected to the root and that the tree has finite height hₙ.

The authors aim to understand the scaling limit of the metric space obtained by dividing all edge lengths by n, i.e. the space Tₙ/n equipped with the uniform probability measure on its vertices. To obtain a limit they impose a series of natural assumptions on the degree‑height profile:

1. Profile convergence (Lim ν). The empirical distribution of heights, given by the normalized profile Dₙ,i/∑_i Dₙ,i, converges weakly to a non‑atomic probability measure ν on