  We consider planar directed last-passage percolation on the square lattice
with general i.i.d. weights and study the geometry of the full set of
semi-infinite geodesics in a typical realization of the random environment. The
structure of the geodesics is studied through the properties of the Busemann
functions viewed as a stochastic process indexed by the asymptotic direction.
Our results are further connected to the ergodic program for and stability
properties of random Hamilton-Jacobi equations. In the exactly solvable
exponential model, our results specialize to give the first complete
characterization of the uniqueness and coalescence structure of the entire
family of semi-infinite geodesics for any model of this type. Furthermore, we
compute statistics of locations of instability, where we discover an unexpected
connection to simple symmetric random walk.
