Geodesics in two-dimensional first-passage percolation
Geodesics in two-dimensional first-passage percolation
We consider standard first-passage percolation on $\mathbb{Z}^2$. Geodesics are nearest-neighbor paths in $\mathbb{Z}^2$, each of whose segments is time-minimizing. We prove part of the conjecture that doubly infinite geodesics do not exist. Our main tool is a result of independent interest about the coalescing of semi-infinite geodesics.