[Paper Review] Connectivity properties of Branching Interlacements

Eviatar B. Procaccia, Yuan Zhang|arXiv (Cornell University)|Dec 1, 2016

Stochastic processes and statistical mechanics5 references3 citations

TL;DR

This paper establishes that in the Branching Interlacements model on $ℤ^d$ for $d \geq 5$, any two vertices visited by the interlacement set are almost surely connected via at most $\lceil d/4\rceil$ conditioned critical branching random walks, and this bound is sharp. The key result is a stochastic dimension analysis showing that connectivity requires exactly $\lceil d/4\rceil$ trajectories, with intersection of such walks occurring precisely when $5 \leq d \leq 8$. The work extends Sznitmann's Random Interlacements framework to branching processes and derives heat kernel bounds for conditioned branching random walks.

ABSTRACT

We consider connectivity properties of the Branching Interlacements model in $\mathbb{Z}^d,~d\ge5$, recently introduced by Angel, Ráth and Zhu in 2016. Using stochastic dimension techniques we show that every two vertices visited by the branching interlacements are connected via at most $\lceil d/4 ceil$ conditioned critical branching random walks from the underlying Poisson process, and that this upper bound is sharp. In particular every such two branching random walks intersect if and only if $5\le d\le 8$. The stochastic dimension of branching random walk result is of independent interest. We additionally obtain heat kernel bounds for branching random walks conditioned on survival.

Motivation & Objective

• To establish connectivity properties of the Branching Interlacements model, a variant of Sznitmann's Random Interlacements built from critical branching random walks.
• To determine the minimal number of conditioned branching random walks required to connect any two vertices almost surely in the interlacement set.
• To prove that the upper bound $\lceil d/4\rceil$ is sharp, meaning $\lceil d/4\rceil - 1$ walks cannot connect all pairs.
• To derive heat kernel bounds for branching random walks conditioned on survival, independent of the main connectivity result.
• To apply stochastic dimension techniques to analyze the connectivity relation $\mathcal{M}$ in the model.

Proposed method

• Use of stochastic dimension theory to analyze the connectivity relation $\mathcal{M}_{u}$ between vertices in the interlacement set.
• Construction of double branching random walks as a representation of critical branching random walks conditioned on survival, using forward and backward components.
• Application of Poisson point process formalism over transient trajectories in $\mathbb{Z}^d$, $d \geq 5$, to model the interlacement process.
• Derivation of upper and lower bounds on the probability of vertex connectivity via $m$ trajectories using tail triviality and independence arguments.
• Use of heat kernel estimates for conditioned branching random walks to support the stochastic dimension analysis.
• Adaptation of techniques from Sznitmann's work on Random Interlacements, particularly the use of disjoint interval conditioning and path dominance arguments.

Experimental results

Research questions

• RQ1 What is the minimal number of conditioned critical branching random walks required to almost surely connect any two vertices in the Branching Interlacement set on $\mathbb{Z}^d$ for $d \geq 5$?
• RQ2 Is the upper bound $\lceil d/4\rceil$ on the number of trajectories needed for connectivity sharp?
• RQ3 For which dimensions $d$ do two conditioned branching random walks almost surely intersect, given they are part of the interlacement process?
• RQ4 What are the heat kernel bounds for branching random walks conditioned on survival, and how do they relate to the connectivity structure?
• RQ5 How does the stochastic dimension of the connectivity relation $\mathcal{M}$ compare to that in the original Random Interlacement model?

Key findings

• Every two vertices in the Branching Interlacement set on $\mathbb{Z}^d$, $d \geq 5$, are almost surely connected via at most $\lceil d/4\rceil$ conditioned critical branching random walks from the underlying Poisson process.
• The bound $\lceil d/4\rceil$ is sharp: there exist pairs of vertices that cannot be connected using only $\lceil d/4\rceil - 1$ such trajectories.
• Two conditioned branching random walks from the interlacement process intersect almost surely if and only if $5 \leq d \leq 8$.
• The stochastic dimension of the connectivity relation $\mathcal{M}$ is 4, which is half the stochastic dimension of the original Random Interlacement model.
• Heat kernel bounds for branching random walks conditioned on survival are derived, supporting the main connectivity analysis.
• The proof relies on tail triviality, independence of disjoint trajectory sets, and conditioning on survival via double branching random walks.