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 l 1 –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 ,2 d ), (iii) superlinear growth for α ∊ (2 d ,2 d + 1), and finally (iv) linear growth for α > 2 d + 1 like the nearest‐neighbor first‐passage percolation model corresponding to α =∞.