Furstenberg–Kesten conjecture on bigeodesics in two-dimensional first-passage percolation

Let $(t_e)$ be i.i.d. edge weights on the nearest-neighbor edges of $\mathbb{Z}^2$, with a continuous distribution. A bigeodesic is a doubly infinite path whose finite segments are geodesics for the induced passage-time metric. Furstenberg–Kesten's bigeodesic conjecture. Almost surely, there are no bigeodesics in dimension two. This is a central question in first-passage percolation; the paper proves the absence of bigeodesics with one end in any deterministic direction under differentiability of the limit-shape boundary, but the unrestricted conjecture remains open.