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 $s\in (d,2d)$, and study the asymptotic of the graph-theoretical (a.k.a. chemical) distance $D(x,y)$ between $x$ and $y$ in the limit as $|x-y|\to\infty$. For the model on $\mathbb Z^d$ we show that, in probability as $|x|\to\infty$, 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 $r\to\infty$ for any nonzero $x\in\mathbb R^d$, 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.