Source: Ann. Probab.
Volume 26, Number 3
We derive the limiting shape for the following model of
first-passage bond percolation on the two-dimensional integer lattice: the
percolation is directed in the sense that admissible paths are nondecreasing in
both coordinate directions. The passage times of horizontal bonds are Bernoulli
distributed, while the passage times of vertical bonds are all equal to a
deterministic constant. To analyze the percolation model, we couple it with a
one-dimensional interacting particle system. This particle process has nonlocal
dynamics in the sense that the movement of any given particle can be influenced
by far-away particles. We prove a law of large numbers for a tagged particle in
this process, and the shape result for the percolation is obtained as a
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