We consider a wide class of ergodic first passage percolation processes on ℤ2 and prove that there exist at least four one-sided geodesics a.s. We also show that coexistence is possible with positive probability in a four-color Richardson’s growth model. This improves earlier results of Häggström and Pemantle [J. Appl. Probab. 35 (1995) 683–692], Garet and Marchand [Ann. Appl. Probab. 15 (2005) 298–330] and Hoffman [Ann. Appl. Probab. 15 (2005) 739–747] who proved that first passage percolation has at least two geodesics and that coexistence is possible in a two-color Richardson’s growth model.
"Geodesics in first passage percolation." Ann. Appl. Probab. 18 (5) 1944 - 1969, October 2008. https://doi.org/10.1214/07-AAP510