Multiple geodesics with the same direction. (English) Zbl 1244.60093

Coupier, David

Electron. Commun. Probab. 16, 517-527 (2011).

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)

Keywords:

geodesics; last-passage percolation; topological end; random tree

Citations:

Zbl 1078.60083; Zbl 1210.60002