Geodesics in first-passage percolation
Daniel Ahlberg (Stockholm University)

In first-passage percolation the edges of the square lattice are equipped with iid non-negative random weights. The weighted graph induces a metric on $\\mathbb\{Z\}^2$, in which the distance between two points corresponds to the minimal weight-sum of a path connecting the points. In the mid 1990s Chuck Newman posed a series of conjectures regarding infinite geodesics in this random metric space, and proved these under an additional condition that remains unverified to this day. Later work has aimed to make rigorous progress on these conjectures, e.g. via the study of coexistence in a model for competing growth. In this talk we shall review parts of this history, and describe versions of Newman's conjectures which we may prove. Finally, as an application of these results we prove two corollaries: First we make precise the relation between geodesics and coexistence in the competing growth model, and second we resolve the ``midpoint problem'' posed by Benjamini, Kalai and Schramm.
[Based on joint work with C. Hoffman]