On Chemical Distance and Local Uniqueness of a Sufficiently Supercritical Finitary Random Interlacement

TL;DR

This paper investigates the geometric structure of the infinite cluster in supercritical Finitary Random Interlacements, showing that the chemical distance aligns with Euclidean distance and establishing local uniqueness of large clusters.

Contribution

It proves the chemical distance is comparable to Euclidean distance and establishes local uniqueness of large clusters in supercritical Finitary Random Interlacements.

Findings

Chemical distance is of the same order as Euclidean distance with high probability.

Shape theorem analogous to Bernoulli percolation and random interlacements.

Local uniqueness of large clusters with high probability.

Abstract

In this paper, we study geometric properties of the unique infinite cluster $\Gamma$ in a sufficiently supercritical Finitary Random Interlacements $\mathcal{FI}^{u,T}$ in $\mathbb{Z}^d, \ d\ge 3$. We prove that the chemical distance in $\Gamma$ is, with stretched exponentially high probability, of the same order as the Euclidean distance in $\mathbb{Z}^d$. This also implies a shape theorem parallel to those for Bernoulli percolation and random interlacements. We also prove local uniqueness of $\mathcal{FI}^{u,T}$, which says any two large clusters in $\mathcal{FI}^{u,T}$ "close to each other" will with stretched exponentially high probability be connected to each other within the same order of the distance between them.