2024/08/29 by Fribergh, Alexander, Hammond, Alan
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2408.16636
We consider supercritical bond percolation in ℤd for d ≥ 3. The origin lies in a finite open cluster with positive probability, and, when it does, the diameter of this cluster has an exponentially decaying tail. For each unit vector \bfℓ, we prove sharp asymptotics for the probability that this cluster contains a vertex x ∈ ℤd that satisfies x ⋅ \bfℓ ≥ u. For an axially aligned \bfℓ, we find this probability to be of the form κexp \ - ζu \(1+ \rm err) for u ∈ ℕ, where \vert \rm err \vert is at most C exp \ - c u1/2 \; for general \bfℓ, the form of the asymptotic depends on whether \bfℓ satisfies a natural lattice condition. To obtain these results, we prove that renewal points in long clusters are abundant, with a renewal block length whose tail is shown to decay as fast as C exp \ - c u1/2 \.