First passage percolation and escape strategies

Consider first passage percolation onℤ dwith passage times given by i.i.d. random variables with common distribution F . Lett π ( u , v ) be the time from u to v for a path π and t ( u , v ) the minimal time among all paths from u to v . We ask whether or not there exist points x , y ∈ ℤ dand a semi‐infinite path π = ( y 0 = y , y 1 , … ) such thatt π ( 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. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 414–423, 2015