On the two-dimensional square lattice, assign i.i.d. nonnegative weights to the

edges with common distribution F. For which F is there an infinite self-avoiding path with

finite total weight? This question arises in first-passage percolation, the study of the

random metric space Z^2 with the induced random graph metric coming from the above edge-weights. It has long been known that there is no such infinite path when F(0)<1/2

(there are only finite paths of zero-weight edges), and there is one when F(0)>1/2 (there

is an infinite path of zero-weight edges). The critical case, F(0)=1/2, is considerably

more difficult due to the presence of finite paths of zero-weight edges on all scales. I will

discuss work with W.-K. Lam and X. Wang in which we give necessary and sufficient

conditions on F for the existence of an infinite finite-weight path. The methods involve

comparing the model to another one, invasion percolation, and showing that geodesics

in first-passage percolation have the same first order travel time as optimal paths in an

embedded invasion cluster.