Graph distances in scale-free percolation: the logarithmic case

Abstract Scale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge with probability depending on independent and identically distributed vertex weights the Euclidean distance $|x-y|$ . Depending various parameters involved, we get rich phase diagram. We study compare it to of vertices. Our main attention regime where distances (poly-)logarithmic in distance. obtain improved bounds logarithmic exponents. light tail regime, correct exponent identified.