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Limit of trees with fixed degree sequence

Arthur Blanc-Renaudie

TL;DR

<3-5 sentence high-level summary> The paper studies scaling limits of uniform rooted trees with fixed degree sequences, showing that after renormalization these discrete trees converge to inhomogeneous continuum random trees (ICRT) under natural moment and tightness conditions. It develops Foata–Fuchs stick-breaking constructions to connect discrete trees to continuum ICRT, proving convergence of the first branches and then GP and GHP convergence for the whole tree, along with sharp height-tail bounds. The framework is extended to 𝒫-trees, random degree sequences, and shows that Lévy trees arise as ICRT with random parameters, unifying several universal limits for random trees. These results provide robust tools for understanding the geometry of critical random graphs and configuration models via continuum limits and have potential applications in related combinatorial structures and stochastic processes.

Abstract

We show, under natural conditions, that uniform rooted trees with fixed degree sequence converge after renormalization toward inhomogeneous continuum random trees (ICRT). We also provide a sharp upper-bound for the tail of their heights. We also extend our results to P-trees, ICRT, and trees with random degree sequence. In passing we confirm a conjecture of Aldous, Miermont, and Pitman stating that Lévy trees are ICRT with random parameters.

Limit of trees with fixed degree sequence

TL;DR

<3-5 sentence high-level summary> The paper studies scaling limits of uniform rooted trees with fixed degree sequences, showing that after renormalization these discrete trees converge to inhomogeneous continuum random trees (ICRT) under natural moment and tightness conditions. It develops Foata–Fuchs stick-breaking constructions to connect discrete trees to continuum ICRT, proving convergence of the first branches and then GP and GHP convergence for the whole tree, along with sharp height-tail bounds. The framework is extended to 𝒫-trees, random degree sequences, and shows that Lévy trees arise as ICRT with random parameters, unifying several universal limits for random trees. These results provide robust tools for understanding the geometry of critical random graphs and configuration models via continuum limits and have potential applications in related combinatorial structures and stochastic processes.

Abstract

We show, under natural conditions, that uniform rooted trees with fixed degree sequence converge after renormalization toward inhomogeneous continuum random trees (ICRT). We also provide a sharp upper-bound for the tail of their heights. We also extend our results to P-trees, ICRT, and trees with random degree sequence. In passing we confirm a conjecture of Aldous, Miermont, and Pitman stating that Lévy trees are ICRT with random parameters.

Paper Structure

This paper contains 27 sections, 38 theorems, 162 equations, 1 figure, 6 algorithms.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Plan of the paper:
  5. Notations:
  6. Convergence of the first branches
  7. Definition of ICRT
  8. Stick-breaking construction for $\mathcal{D}$-trees
  9. Preliminary: continuum modification of $\Upsilon^\mathcal{D}$
  10. Convergence of $\mathcal{D}$-trees toward ICRT: proof of Proposition \ref{['pro:First_branches_D']}
  11. Gromov--Prokhorov convergence of $\mathcal{D}$-trees
  12. Preliminary: vertices with small degree behaves like leaves
  13. GP convergence of $\mathcal{D}$-trees toward ICRT: proof of Theorem \ref{['D_GP_T']}
  14. Gromov--Hausdorff--Prokhorov convergence and height of $\mathcal{D}$-trees
  15. Preliminaries: technical results on $\tilde{\mu}$ and $\mu$
  16. ...and 12 more sections

Key Result

Theorem 1.1

If ${\mathcal{D}^n}\Rightarrow \Theta$, ${\mathfrak p}^n\to 0$ uniformly, and eq:ThetaSet holds, then the following convergence holds for the weak Gromov--Prokhorov (GP) topology (see Appendix GPdef for definition of the topology):

Figures (1)

  • Figure 1: Stick breaking construction of a $\mathcal{D}$-tree with $\mathcal{D}=(1,2,1,3,3,0,0,\dots)$ and $(A^\mathcal{D}_i)_{1\leq i \leq \aleph-1}=(V_4,V_5,V_2 ,V_5,V_3,V_4,V_5,V_4,V_1,V_2)$. The exploration starts at $V_4$ then follows the white-black arrow toward $L_1$, then jumps at $V_5$ to follow the path toward $L_2$ and so on… Here $Y_1=4, Y_2=6, Y_3=7\dots$, $Z_1=2, Z_2=1, Z_3=2\dots$, $X_1=9, X_2=10, X_3=5\dots$

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Proposition 3.2 from ICRT1
  • Theorem 2.2: Theorem 3.3 from ICRT1
  • Proposition 2.3
  • Remark
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 67 more