Daniel Ahlberg: Random coalescing geodesics in first-passage percolation

Abstract:
Since the work of Kardar-Parisi-Zhang in the mid 1980s, it has been widely believed that a large class of two-dimensional growth models should obey the same asymptotic behaviour. To rigorously understand the predictions of KPZ-theory has since been one of the most central themes in mathematical physics. One prominent model believed to belong to this class is known as first-passage percolation. It can be interpreted as the random metric on Z^2 obtained by assigning non-negative i.i.d. weights to the edges of the nearest neighbour lattice. We shall discuss properties of geodesics in this metric and their connection to KPZ-theory. We develop an ergodic theory for infinite geodesics via the study of what we shall call `random coalescing geodesics’. As an application of this theory we answer a question posed by Benjamini, Kalai and Schramm in 2003, that has come to be known as the `midpoint problem’. This is joint work with Chris Hoffman.