Analysis and Mathematical Physics

We study the probability that a geodesic passes through a prescribed edge in first-passage percolation on Z^d. Benjamini, Kalai and Schramm conjectured that this probability tends to zero as the length of the geodesic tends to infinity, as long as the edge is not too close to the endpoints of the geodesic. I will present a short proof that this probability is arbitrarily small for all edges except for constantly many of them. I will also discuss the connection of this problem to the phenomenon of coalescence of geodesics.

This is a joint work with Ron Peled and Barbara Dembin.