Geodesics in first passage percolation

Abstract

We consider a wide class of ergodic first passage percolation processes on ℤ 2 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. © Institute of Mathematical. Statistics, 2008.