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

2020/09/09 by Cai, Zhenhao, Han, Xiao, Ye, Jiayan +1
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2009.04044

Abstract

In this paper, we study geometric properties of the unique infinite cluster Γ in a sufficiently supercritical Finitary Random Interlacements FI^u,T in ℤ^d, d≥ 3. We prove that the chemical distance in Γ is, with stretched exponentially high probability, of the same order as the Euclidean distance in ℤ^d. This also implies a shape theorem parallel to those for Bernoulli percolation and random interlacements. We also prove local uniqueness of FI^u,T, which says any two large clusters in 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.

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