Multiple phase transitions in long-range first-passage percolation on square lattices

Abstract

We consider a model of long-range first-passage percolation on the $d$ dimensional square lattice $Z^d$ in which any two distinct vertices $x,y\in Z^d$ are connected by an edge having exponentially distributed passage time with mean $||x-y|{|}^{\alpha +o(1)}$, where $\alpha >0$ is a fixed parameter and $||\cdot ||$ is the ${\ell }_1$-norm on $Z^d$. We analyze the asymptotic growth rate of the set $B_t$, which consists of all $x\in Z^d$ such that the first-passage time between the origin 0 and $x$ is at most $t$, as $t\to \infty$. We show that depending on the values of $\alpha$ there are four growth regimes: (i) instantaneous growth for $\alpha <d$, (ii) stretched exponential growth for $\alpha \in (d,2d)$, (iii) superlinear growth for $\alpha \in (2d,2d+1)$ and finally (iv) linear growth for $\alpha >2d+1$ like the nearest-neighbor first-passage percolation model corresponding to $\alpha =\infty$.

Final version, 46 pages; 6 figures. To appear in CPAM