Sharp asymptotic for the chemical distance in long-range percolation

Abstract

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 $(\log r{)}^{\hat{I}}”$, where $\hat{I}”:=1/{\log }_2(1/{\hat{I}}^3)$ for ${\hat{I}}^3:=s/(2d)$. 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 $\ddot{I}•(r)(\log r{)}^{\hat{I}}”$ for $\ddot{I}•$ a positive, continuous (deterministic) function obeying $\ddot{I}•(r^3)=\ddot{I}•(r)$ 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.

22 pages, 2 figs