Explosion and distances in scale-free percolation

Abstract

We investigate the weighted scale-free percolation (SFPW) model on $\mathbb Z^d$. In the SFPW model, the vertices of $\mathbb Z^d$ are assigned i.i.d. weights $(W_x)_{x\in \mathbb Z^d}$, following a power-law distribution with tail exponent $\tau>1$. Conditioned on the collection of weights, the edges $(x,y)_{x, y \in \mathbb Z^d}$ are present independently with probability that a Poisson random variable with parameter $\lambda W_x W_y/(\|x-y\|)^\alpha$ is at least one, for some $\alpha, \lambda>0$, and where $\|\cdot \|$ denotes the Euclidean distance. After the graph is constructed this way, we assign independent and identically distributed (i.i.d.) random edge lengths from distribution $L$ to all existing edges in the graph. The focus of the paper is to determine when is the obtained model \emph{explosive}, that is, when it is possible to reach infinitely many vertices in finite time from a vertex. We show that explosion happens precisely for those edge length distributions that produce explosive branching processes with infinite mean power law offspring distributions. For non-explosive edge-length distributions, when $\gamma \in (1,2)$, we characterise the asymptotic behaviour of the time it takes to reach the first vertex that is graph distance $n$ away. For $\gamma>2$, we show that the number of vertices reachable by time $t$ from the origin grows at most exponentially, thus explosion is never possible. For the non-explosive edge-length distributions, when $\gamma \in (1,2)$, we further determine the first order asymptotics of distances when $\gamma\in (1,2)$. As a corollary we obtain a sharp upper and lower bound for graph distances, closing a gap between a lower and upper bound on graph distances in Deijfen Hofstad `13 when $\gamma \in(1,2), \tau>2$.