LA Probability Forum

First-passage percolation on the square lattice is a random growth model in which each edge of Z^2 is assigned an i.i.d. nonnegative weight.  The passage time between two points is the smallest total weight of a nearest-neighbor path connecting them, and a path achieving this minimum is called a geodesic.  Typically, the number of edges in a geodesic is comparable to the Euclidean distance between its endpoints.  However, when the edge-weights take the value 0 with probability 1/2, a strikingly different behavior occurs: geodesics travel primarily on critical clusters of zero-weight edges, whose internal graph distance scales superlinearly with Euclidean distance.  Determining the precise degree of this superlinear scaling is a challenging and ongoing endeavor.  I will discuss recent progress on this front (joint work with David Harper, Evan Sorensen, and Xiao Shen), along with complementary results on a dual problem, where we restrict path lengths and analyze passage times (joint with Jack Hanson and Daniel Slonim).