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October 2018 The size of the boundary in first-passage percolation
Michael Damron, Jack Hanson, Wai-Kit Lam
Ann. Appl. Probab. 28(5): 3184-3214 (October 2018). DOI: 10.1214/18-AAP1388

Abstract

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 $t$.

Citation

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Michael Damron. Jack Hanson. Wai-Kit Lam. "The size of the boundary in first-passage percolation." Ann. Appl. Probab. 28 (5) 3184 - 3214, October 2018. https://doi.org/10.1214/18-AAP1388

Information

Received: 1 September 2017; Revised: 1 January 2018; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974777
MathSciNet: MR3847985
Digital Object Identifier: 10.1214/18-AAP1388

Subjects:
Primary: 60K35, 82B43

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 5 • October 2018
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