Double Behavior of Critical First-Passage Percolation

Abstract

We consider standard first passage percolation on the Z ^d lattice. Let {x(e) : e an edge of Z ^d} be an i.i.d. family of random variables with distribution F. Denote by c _0,n the first passage time from the origin to the boundary of [−n, n]^d. For d = 2 we show that there exist two curves F _a and G _b both with F _a (0) = G _b (0) = p _c such that lim _n→∞ E c _0,n exists whenever F(0) = p _c and F ≥ F _a or blows up whenever F(0) = p _c and F ≤ G _b , respectively. We also can obtain the corresponding results for the passage times a _0,n and b _0,n . Furthermore, we will investigate the behavior of lim_n→∞ E(c _0,n ) when F(0) is near p _c . For a large d, we show the lower bound for E a _0,n is larger than C log log n for some constant C > 0, and discuss the existence of routes for a _0,n and b _0,n when F(0) = p _c .