Multiple Phase Transitions in Long-Range First-Passage Percolation on Square Lattices

We consider a model of long-range first-passage percolation on the d-dimensional square lattice Z^d in which any two distinct vertices x,y∈Z^d are connected by an edge having exponentially distributed passage time with mean x-y ^α+o(1), where α>0 is a fixed parameter and . is the l¹-norm on Z^d. We analyze the asymptotic growth rate of the set ß_t, which consists of all x∈Z^d such that the first-passage time between the origin 0 and x is at most t as t→∞. We show that depending on the values of α there are four growth regimes: (i) instantaneous growth for α<^d, (ii) stretched exponential growth for α∈(d,2d), (iii) superlinear growth for α∈(2d,2d+1), and finally (iv) linear growth for α>2d+1 like the nearest-neighbor first-passage percolation model corresponding to α=∞.