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• W2623592616 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 $(W_x)_{xin mathbb Z^d}$, 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 $gammain (1,2)$. 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 $gamma in(1,2), tau>2$." @default.
• W2623592616 created "2017-06-15" @default.
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• W2623592616 date "2017-06-08" @default.
• W2623592616 modified "2023-09-27" @default.
• W2623592616 title "Explosion and distances in scale-free percolation" @default.
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