Multipoint connectivity in the branching interlacement process

TL;DR

This paper investigates the connectivity properties of the branching interlacement model, establishing sharp bounds on the number of trajectories needed to connect multiple points, extending previous results.

Contribution

It provides new sharp bounds on the number of trajectories required for connecting points in the branching interlacement, generalizing to multiple points.

Findings

Two points are connected by at most eil(d/4) trajectories.

The bounds are sharp; some points are not connected by fewer trajectories.

The results extend to connect multiple points with explicit bounds.

Abstract

We consider the branching interlacement model introduced by Zhu as an analog of Sznitman's random interlacement for branching random walks. We show that two points of the interlacement are connected via at most $\lceil d/4 \rceil$ trajectories of the interlacement, using a different proof than Procaccia and Zhang. This upper bound is sharp, in the sense that almost surely there exist two points not connected by $\lceil d/4\rceil - 1$ trajectories. We extend this result by proving that $k$ points of the interlacement are connected via at most $\lceil d(k-1)/4\rceil -(k-2)$ trajectories, and that this bound is also sharp.