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The size of the boundary in first-passage percolation. (English) Zbl 1402.60127
Summary: First-passage percolation is a random growth model defined using i.i.d. edge-weights $$(t_{e})$$ on the nearest-neighbor edges of $$\mathbb{Z}^{d}$$. An initial infection occupies the origin and spreads along the edges, taking time $$t_{e}$$ to cross the edge $$e$$. In this paper, we study the size of the boundary of the infected (“wet”) region at time $$t$$, $$B(t)$$. It is known that $$B(t)$$ grows linearly, so its boundary $$\partial B(t)$$ has size between $$ct^{d-1}$$ and $$Ct^{d}$$. Under a weak moment condition on the weights, we show that for most times, $$\partial B(t)$$ has size of order $$t^{d-1}$$ (smooth). On the other hand, for heavy-tailed distributions, $$B(t)$$ contains many small holes, and consequently we show that $$\partial B(t)$$ has size of order $$t^{d-1+\alpha }$$ for some $$\alpha >0$$ depending on the distribution. In all cases, we show that the exterior boundary of $$B(t)$$ [edges touching the unbounded component of the complement of $$B(t)$$] is smooth for most times. Under the unproven assumption of uniformly positive curvature on the limit shape for $$B(t)$$, we show the inequality $$\#\partial B(t)\leq (\log t)^{C}t^{d-1}$$ for all large $$t$$.
MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
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