We introduce a simplified model of planar first passage percolation where weights along vertical edges are deterministic. We show that the limit shape has a flat edge in the vertical direction if and only if the random distribution of the horizontal edges has an atom at the infimum of its support. Furthermore, we present bounds on the upper and lower derivative of the time constant.
Random growth models: shape and convergence rate
Differentiability of limit shapes in continuous first passage percolation models
Properties of the limit shape for some last passage growth models in random environments (Dissertation)
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