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Non triviality of the percolation threshold and Gumbel fluctuations for Branching Interlacements

Bruno Schapira

TL;DR

This work establishes that the vacant set of Branching Interlacements undergoes a nontrivial percolation transition in dimensions d \ge 5, proving 0 < u_* < \infty. A central technical contribution is a sharp decorrelation inequality for events depending on well-separated boxes, adapted to the tree-indexed branching random walk framework and the branching capacity. The authors also derive Gumbel fluctuations for the cover level M(K) of finite sets, paralleling known results for Random Interlacements, and provide a robust probabilistic toolkit (including hitting probabilities and hierarchical box arguments) essential for analyzing correlated fields arising from branching structures. Together, these results deepen the understanding of how critical branching dynamics shape percolation phenomena and finite-set cover times in high-dimensional lattices, with potential applications to related branching-structure models.

Abstract

We consider the model of Branching Interlacements, introduced by Zhu, which is a natural analogue of Sznitman's Random Interlacements model, where the random walk trajectories are replaced by ranges of some suitable tree-indexed random walks. We first prove a basic decorrelation inequality for events depending on the state of the field on distinct boxes. We then show that in all relevant dimensions, the vacant set undergoes a nontrivial phase transition regarding the existence of an infinite connected component. Finally we obtain the Gumbel fluctuations for the cover level of finite sets, which is analogous to Belius' result in the setting of Random Interlacements.

Non triviality of the percolation threshold and Gumbel fluctuations for Branching Interlacements

TL;DR

This work establishes that the vacant set of Branching Interlacements undergoes a nontrivial percolation transition in dimensions d \ge 5, proving 0 < u_* < \infty. A central technical contribution is a sharp decorrelation inequality for events depending on well-separated boxes, adapted to the tree-indexed branching random walk framework and the branching capacity. The authors also derive Gumbel fluctuations for the cover level M(K) of finite sets, paralleling known results for Random Interlacements, and provide a robust probabilistic toolkit (including hitting probabilities and hierarchical box arguments) essential for analyzing correlated fields arising from branching structures. Together, these results deepen the understanding of how critical branching dynamics shape percolation phenomena and finite-set cover times in high-dimensional lattices, with potential applications to related branching-structure models.

Abstract

We consider the model of Branching Interlacements, introduced by Zhu, which is a natural analogue of Sznitman's Random Interlacements model, where the random walk trajectories are replaced by ranges of some suitable tree-indexed random walks. We first prove a basic decorrelation inequality for events depending on the state of the field on distinct boxes. We then show that in all relevant dimensions, the vacant set undergoes a nontrivial phase transition regarding the existence of an infinite connected component. Finally we obtain the Gumbel fluctuations for the cover level of finite sets, which is analogous to Belius' result in the setting of Random Interlacements.

Paper Structure

This paper contains 14 sections, 25 theorems, 150 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notation
  4. The infinite invariant tree
  5. Tree-indexed random walk, Green's function, and Branching capacity
  6. Branching interlacements point process
  7. Hitting probabilities for branching random walks
  8. Proof of Proposition \ref{['prop.cov']}
  9. Proof of Theorem \ref{['thm.cov']}
  10. Proof of Theorem \ref{['thm:pt']}
  11. Preliminaries and notation
  12. Proof of $u_*<\infty$
  13. Proof of $u_*>0$
  14. Proof of Theorem \ref{['thm.cover']}

Key Result

Proposition 1.1

There exists a constant $c>0$, such that for any $u>0$,

Figures (1)

  • Figure 1: An infinite tree rooted at $0$, seen from $-8$ and relabelled.

Theorems & Definitions (43)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2: Zhu16
  • Corollary 2.3
  • ...and 33 more