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Multiple geodesics with the same direction. (English) Zbl 1244.60093
Summary: 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 P. A. Ferrari and L. P. R. Pimentel [Ann. Probab. 33, No. 4, 1235–1254 (2005; Zbl 1078.60083)]. In particular, using a recent result by G. F. Lawler and V. Limic [Random walk: a modern introduction. Cambridge: Cambridge University Press (2010; Zbl 1210.60002)] and a local modification argument, we prove there is no (random) direction with more than two geodesics with probability 1.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
05C80 Random graphs (graph-theoretic aspects)
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