The directed last-passage percolation (LPP) model with independent exponential times is considered. We complete the study of asymptotic directions of infinite geodesics, started by Ferrari and Pimentel . In particular, using a recent result of  and a local modification argument, we prove there is no (random) direction with more than two geodesics with probability 1.
"Multiple geodesics with the same direction." Electron. Commun. Probab. 16 517 - 527, 2011. https://doi.org/10.1214/ECP.v16-1656