Abstract:
Consider first passage percolation on Z^d with passage times given by i.i.d. random variables with common distribution F. Let t_\pi(u,v) be the time from u to v for a path \pi and t(u,v) the minimal time among all paths from u to v. We ask whether or not there exist points x,y \in Z^d and a semi-infinite path \pi=(y_0=y,y_1,\dots) such that t_\pi(y,y_{n+1})<t(x,y_n) for all n. Necessary and sufficient conditions on F are given for this to occur. When the support of F is unbounded, we also obtain results on the number of edges with large passage time used by geodesics.