THE SIZE OF THE BOUNDARY IN FIRST-PASSAGE PERCOLATION

First-passage percolation is a random growth model defined using i.i.d. edge-weights (t(e)) on the nearest-neighbor edges of 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 derivative 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 derivative 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 derivative 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 derivative B(t) <= (log t) C-t(d-1) for all large t.