Uniform attachment with freezing: Scaling limits

TL;DR

This paper studies the scaling limits of a variant of random recursive trees with freezing, revealing a phase transition and a critical regime where the limit is a random compact real tree related to coalescent processes.

Contribution

It introduces and analyzes a new model of uniform attachment with freezing, identifying phase transitions and the nature of scaling limits, including a critical regime with condensation phenomena.

Findings

Identified a phase transition in the model.

Discovered a critical regime with a real tree limit.

Observed condensation phenomena in the critical regime.

Abstract

We investigate scaling limits of trees built by uniform attachment with freezing, which is a variant of the classical model of random recursive trees introduced in a companion paper. Here vertices are allowed to freeze, and arriving vertices cannot be attached to already frozen ones. We identify a phase transition when the number of non-frozen vertices roughly evolves as the total number of vertices to a given power. In particular, we observe a critical regime where the scaling limit is a random compact real tree, closely related to a time non-homogenous Kingman coalescent process identified by Aldous. Interestingly, in this critical regime, a condensation phenomenon can occur.