Sharp asymptotic for the chemical distance in long‐range percolation

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 ∈ (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|→∞. For the model on Zd we show that, in probability as |x|→∞, the distance D(0,x) is squeezed between two positive multiples of (logr)Δ, where Δ:=1/log2(1/γ) for γ: = s/(2d). For the model on Rd we show that D(0,xr) is, in probability as r→∞ for any nonzero x∈Rd, asymptotic to φ(r)(logr)Δ for φ a positive, continuous (deterministic) function obeying φ(rγ) = φ(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.