Sharp asymptotic for the chemical distance in long-range percolation

We consider instances of long-range percolation on $mathbb Z^d$ and $mathbb R^d$, where points at distance $r$ get connected by an edge with probability proportional to $r^{-s}$, for $sin (d,2d)$, and study the asymptotic of the graph-theoretical (a.k.a. chemical) distance $D(x,y)$ between $x$ and $y$ in the limit as $|x-y|toinfty$. For the model on $mathbb Z^d$ we show that, in probability as $|x|toinfty$, the distance $D(0,x)$ is squeezed between two positive multiples of $(log r)^Delta$, where $Delta:=1/log_2(1/gamma)$ for $gamma:=s/(2d)$. For the model on $mathbb R^d$ we show that $D(0,xr)$ is, in probability as $rtoinfty$ for any nonzero $xinmathbb R^d$, asymptotic to $phi(r)(log r)^Delta$ for $phi$ a positive, continuous (deterministic) function obeying $phi(r^gamma)=phi(r)$ for all $r>1$. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly-exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain.