Abstract:

 I shall discuss first passage percolation on Cayley graphs of Gromov hyperbolic groups, i.e., the random metric obtained by assigning independent and identically distributed positive random weights to the edges of the graph. Under mild conditions on the weights, I shall discuss several results about the structure of semi-infinite and bi-infinite geodesics in the random metric and in particular their connections to the topological properties of the Gromov boundary. I shall also briefly discuss the similarity and contrasts of the results in the hyperbolic background geometry to what is expected in first passage percolation on Euclidean lattices.