The Annals of Applied Probability

Geodesics in first passage percolation

Christopher Hoffman

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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.

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Ann. Appl. Probab., Volume 18, Number 5 (2008), 1944-1969.

First available in Project Euclid: 30 October 2008

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

First passage percolation Richardson’s growth model


Hoffman, Christopher. Geodesics in first passage percolation. Ann. Appl. Probab. 18 (2008), no. 5, 1944--1969. doi:10.1214/07-AAP510.

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