Limiting shape for directed percolation models

@article{Martin2004LimitingSF,
  title={Limiting shape for directed percolation models},
  author={James B. Martin},
  journal={Annals of Probability},
  year={2004},
  volume={32},
  pages={2908-2937}
}
We consider directed first-passage and last-passage percolation on the nonnegative lattice Z d + , d ≥ 2, with i.i.d. weights at the vertices. Under certain moment conditions on the common distribution of the weights, the limits g(x) = lim n→ ∞ n- 1 T(|nx|) exist and are constant a.s. for x ∈ R d + , where T(z) is the passage time from the origin to the vertex z ∈ Z d + . We show that this shape function g is continuous on R d + , in particular at the boundaries. In two dimensions, we give more… 

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