  In first-passage percolation, we place i.i.d. continuous weights at the edges
of Z^2 and consider the weighted graph metric. A distance-minimizing path
between points x and y is called a geodesic, and a bigeodesic is a
doubly-infinite path whose segments are geodesics. It is a famous conjecture
that almost surely, there are no bigeodesics. In the '90s, Licea-Newman showed
that, under a curvature assumption on the "asymptotic shape", all infinite
geodesics have an asymptotic direction, and there is a full-measure set D of
[0, 2 pi) such that for any theta in D, there are no bigeodesics with one end
directed in direction theta. In this paper, we show that there are no
bigeodesics with one end directed in any deterministic direction, assuming the
shape boundary is differentiable. This rules out existence of ground state
pairs for the related disordered ferromagnet whose interface has a
deterministic direction. Furthermore, it resolves the Benjamini-Kalai-Schramm
"midpoint problem" under the extra assumption that the limit shape boundary is
differentiable.
