The Annals of Applied Probability

The size of the boundary in first-passage percolation

Michael Damron, Jack Hanson, and Wai-Kit Lam

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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 $t$.

Article information

Ann. Appl. Probab., Volume 28, Number 5 (2018), 3184-3214.

Received: September 2017
Revised: January 2018
First available in Project Euclid: 28 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

First-passage percolation boundary Eden model limit shape


Damron, Michael; Hanson, Jack; Lam, Wai-Kit. The size of the boundary in first-passage percolation. Ann. Appl. Probab. 28 (2018), no. 5, 3184--3214. doi:10.1214/18-AAP1388.

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