Multi-type Shape Theorems for Fpp Models

In standard planar first-passage percolation [11] each pair x and y of nearest-neighbor of Z has an edge connecting them and each edge is equipped with a non-negative random variable (passage time) which may be interpreted as the time it takes for an infection to be transmitted from x to y. We assume these random variables are i.i.d. with a continuous distribution F. The passage time t(γ) for a nearest-neighbor path γ is simply the sum of the passage times along the path. For x,y ∈ Z, the first-passage time from x to y, which we denote T (x,y), is the infimum of t(γ) over all paths γ from x to y. For t ≥ 0, let B(t) be the set of sites x reached from the origin 0 by time t, i.e. T (0,x) ≤ t. One may think of sites in x ∈ B(t) as infected and those in B(t) as healthy, and that at time 0 the origin 0 is infected by some type of disease. The process {