First Passage Percolation and Escape Strategies

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 is an element of Z(d) and a semi-infinite path pi = (y(0) = y,y1, ...) 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. (c) 2014 Wiley Periodicals, Inc.