Abstrakt
We consider instances of long-range percolation on Zd and Rd, where points at distance r get connected by an edge with probability proportional to r(-s), for s is an element of (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|->infinity. For the model on Zd we show that, in probability as |x|->infinity, the distance D(0,x) is squeezed between two positive multiples of (logr)Delta, where Delta:=1/log2(1/gamma) for gamma: = s/(2d).
For the model on Rd we show that D(0,xr) is, in probability as r ->infinity for any nonzero x is an element of Rd, asymptotic to phi(r)(logr)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.