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Limiting shape for directed percolation models. (English) Zbl 1065.60149

The author considers a class of first- and last-passage percolation models on orthants. Under rather weak moment conditions a shape theorem is proved, and continuity of the shape also at the boundaries, as well as the asymptotics in \(d= 2\) at the boundary are found. Also the possibilities are discussed when the moment conditions of the theorem are not fulfilled, supported by numerical simulations.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C43 Time-dependent percolation in statistical mechanics
82B43 Percolation
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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