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−1T(⌊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 precise asymptotics for the behavior of g near the boundaries; these asymptotics depend on the common weight distribution only through its mean and variance. In addition we discuss growth models which are naturally associated to the percolation processes, giving a shape theorem and illustrating various possible types of behavior with output from simulations.
"Limiting shape for directed percolation models." Ann. Probab. 32 (4) 2908 - 2937, October 2004. https://doi.org/10.1214/009117904000000838