Uniform attachment with freezing

TL;DR

This work introduces Uniform attachment with freezing, a dynamic where existing vertices may freeze and prevent future attachments, governed by a deterministic sequence of steps encoded in $\boldsymbol{x}$. It develops two equivalent constructions (a forward recursive process and a backward growth-coalescent forest) and establishes local-limit results, linking the model to Bienaymé trees with geometric offspring in a precise regime. The authors derive detailed height results, proving concentration around a natural height scale $\mathsf{h}_n^+$ and obtaining sharp log-scale limits in a linear-active regime, along with distance asymptotics between random vertices. They further apply the framework to the geometry of infection trees in stochastic SIR dynamics, showing convergence to Bienaymé-type limits under scaling and discussing extensions and open problems. Overall, the paper unifies growth-coalescent techniques, local limits, and height/distance analyses to illuminate the impact of freezing on the geometry of dynamically built random trees and their epidemiological analogs.

Abstract

In the classical model of random recursive trees, trees are recursively built by attaching new vertices to old ones. What happens if vertices are allowed to freeze, in the sense that new vertices cannot be attached to already frozen ones? We are interested in the impact of freezing on the height of such trees.

Figures (10)

• Figure 1: Simulation of a tree of size $10000$ built by uniform attachment with freezing, when the number of active (i. e. non-frozen) vertices roughly evolves as a positive fraction of the total number of vertices. Frozen vertices are blue; active vertices are red. The animation (played with Acrobat Reader) shows the resulting process as the number of steps increases.
• Figure 2: Plot of the function $(\frac{{c+1} }{2c}f(c): 0 <c \leq 1)$.
• Figure 3: On the left is represented the walk $(S_n(\mathrm{\mathbf{x}}))_{n \geq 0}$ up to time $n=5$ associated with the sequence $\mathrm{\mathbf{x}}=+1,-1,+1,+1,-1,\dots$. On the right, a possible realisation of the trees $\mathcal{T}_0(\mathrm{\mathbf{x}})$ to $\mathcal{T}_5(\mathrm{\mathbf{x}})$ given this sequence. Frozen vertices have been colored in blue.
• Figure 4: An illustration of Algorithm \ref{['algo2']} with $n=5$ and $(\mathrm x_{5},\mathrm x_{4},\mathrm x_{3},\mathrm x_{2},\mathrm x_{1})=(-1,1,1,-1,1)$ (this is the same sequence as in Fig. \ref{['fig exemple arbre recursif avec gel']}). For example, since $\mathrm x_{2}=-1$, $\mathcal{F}^{5}_{1}$ is obtained from $\mathcal{F}^{5}_{2}$ by adding a new tree made of a vertex labeled $2$. Since $\mathrm x_{4}=1$, to build $\mathcal{F}^{5}_{3}$ from $\mathcal{F}^{5}_{4}$ we have chosen in $\mathcal{F}^{5}_{3}$ two trees $(T_{1},T_{2})$ with $T_{1}$ being the vertex $a_{1}$ and $T_{2}$ being the vertex $a_{2}$, and we have added an edge labelled $4$ between the roots $r(T_{1})=a_{1}$ and $r(T_{2})=a_{2}$ of respectively$T_{1}$ and $T_{2}$, and rooted the tree thus obtained at $r(T_{1})=a_{1}$.
• Figure 5: Simulation of Algorithm \ref{['algo2']}: the animation (played with Acrobat Reader) shows the resulting process as the number of steps increases. When a new vertex appears, its color is chosen at random. When two trees merge, the resulting tree keeps the color of the largest of the two trees.

Theorems & Definitions (46)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• Lemma 6
• proof
• Lemma 7
• proof
• Theorem 8