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The scaling limit of the root component in the Wired Minimal Spanning Forest of the Poisson Weighted Infinite Tree

Omer Angel, Delphin Sénizergues

Abstract

In this paper we prove a scaling limit result for the component of the root in the Wired Minimal Spanning Forest (WMSF) of the Poisson-Weighted Infinite Tree (PWIT), where the latter tree arises as the local weak limit of the Minimal Spanning Tree (MST) on the complete graph endowed with i.i.d. weights on its edges. The limiting object can be obtained by aggregating independent Brownian trees using two types of gluing procedures: one that we call the Brownian tree aggregation process and resembles the so-called stick-breaking construction of the Brownian tree; and another one that we call the chain construction, which simply corresponds to gluing a sequence of metric spaces along a line.

The scaling limit of the root component in the Wired Minimal Spanning Forest of the Poisson Weighted Infinite Tree

Abstract

In this paper we prove a scaling limit result for the component of the root in the Wired Minimal Spanning Forest (WMSF) of the Poisson-Weighted Infinite Tree (PWIT), where the latter tree arises as the local weak limit of the Minimal Spanning Tree (MST) on the complete graph endowed with i.i.d. weights on its edges. The limiting object can be obtained by aggregating independent Brownian trees using two types of gluing procedures: one that we call the Brownian tree aggregation process and resembles the so-called stick-breaking construction of the Brownian tree; and another one that we call the chain construction, which simply corresponds to gluing a sequence of metric spaces along a line.
Paper Structure (77 sections, 30 theorems, 244 equations, 3 figures, 2 tables)

This paper contains 77 sections, 30 theorems, 244 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The scaling limit of $M_n$.
  3. The local weak limit.
  4. Towards a non-compact scaling limit.
  5. Structure of the root component of the WMSF of the PWIT
  6. Invasion percolation, forward maximal edges and ponds.
  7. Invasion percolation from every vertex.
  8. Summary of the results
  9. Joint construction of $\mathcal{T}^\infty$ and $\mathcal{M}^\infty$.
  10. Convergence results.
  11. Construction of $\mathcal{M}^\bullet$ from $\mathcal{T}^\bullet$: the Brownian tree aggregation process.
  12. Organization of this paper
  13. Constructions and convergences
  14. Reminder about the pointed GH and GHP topologies and local versions
  15. Gromov-Hausdorff-Prokhorov metric for compact spaces
  16. ...and 62 more sections

Key Result

theorem 1

Seen as a measured metric space, we have the following convergence in the Gromov--Hausdorff--Prokhorov topology where the limiting object $\mathcal{M}^{\mathrm{comp}}$ is a compact random tree that has Minkowski dimension equal to $3$ almost surely.

Figures (3)

  • Figure 1: The objects $M_n$, $\mathcal{M}^{\mathrm{comp}}$, $\mathrm{M}^\infty$, and $\mathcal{M}^\infty$ and their relationships. This paper focuses on the red arrow. The solid black arrows are already proved in the literature, and the dashed ones are conjectured.
  • Figure 2: Structure of invasion percolation clusters on the PWIT
  • Figure 3: Construction of the non-compact continuous object $\mathcal{M}^{\infty}$ (on the bottom) from a Poisson point process $\mathscr{P}$ (on top). The color of the trees below correspond to the atoms of the PPP above. Note that in $\mathscr{P}$ there is an accumulation of atoms near $\infty$ (not represented here).

Theorems & Definitions (63)

  • theorem 1: Theorem 1.1 of addario-berry_scaling_2017
  • theorem 2: aldous_objective_2004addario-berry_local_2013nachmias_wired_2024
  • theorem 3
  • theorem 4
  • Definition 5
  • theorem 6: Theorem 2.5 of abraham_note_2013
  • theorem 7: Theorem 2.9 and Proposition 2.10 of abraham_note_2013
  • Lemma 8
  • Remark 9
  • proposition 1
  • ...and 53 more