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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 \(t\).
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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