Graph distances in scale-free percolation: the logarithmic case

Scale-free percolation is a spatial random graph model with vertex set $mathbb{Z}^d$. Vertices $x$ and $y$ are connected with probability depending on i.i.d. vertex weights and the Euclidean distance. Depending on the various parameters involved, we get a rich phase diagram. We study graph distances (in comparison to Euclidean distances). Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.