The size of the boundary in first-passage percolation

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 $t.$