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Christopher Hoffman : Geodesics in first passage percolation

First passage percolation is the study of a random metric space generated by replacing each edge in a graph by an edge of a random length. The distance between two vertices u and v is the length of the shortest path connecting u and v. An infinite path P is a geodesic if for any two vertices u and v on P the shortest path between them in the random graph is along P. It is easy to show that in the nearest neighbor graph with vertices Z^2 that there exists at least one (one sided) infinite geodesic starting at any given vertex. It is widely expected that there are infinitely many such one sided infinite geodesics that begin at the origin, with (at least) one in every direction. But it turns out to be very difficult to prove that there are even two with positive probability. We will discuss some recent results which get closer to proving this widely held belief.

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