First passage percolation and escape strategies1. Issue 3 (23rd May 2014) - Andjel, Enrique D.; Vares, Maria E.
Abstract Consider first passage percolation on <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trgrq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ℤd</alternatives> with passage times given by i.i.d. random variables with common distribution F. Let <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trgs8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />tπ(u, v)</alternatives> be the time from u to v for a path π and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trgkz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />t(u, v)</alternatives> the minimal time among all paths from u to v. We ask whether or not there exist points <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trgn2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />x, y∈ℤd</alternatives> and a semi‐infinite path <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trgg9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />π=(y0=y, y1, …)</alternatives> such that <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trghv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />tπ(y, yn+1)<t(x, yn)</alternatives> 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