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

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