Sharp asymptotic for the chemical distance in long‐range percolation. Issue 3 (4th April 2019) - Biskup, Marek; Lin, Jeffrey
Abstract : We consider instances of long‐range percolation on Z d and R d, where points at distance r get connected by an edge with probability proportional to r − s, for s  ∈ ( d, 2 d ), 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 Z d we show that, in probability as | x |→ ∞, the distance D (0, x ) is squeezed between two positive multiples of ( log r ) Δ, where Δ : = 1 / log 2 ( 1 / γ ) for γ : =  s /(2 d ). For the model on R d we show that D (0, xr ) is, in probability as r → ∞ for any nonzero x ∈ R d, asymptotic to φ ( r ) ( log r ) Δ for φ a positive, continuous (deterministic) function obeying φ ( r γ ) =  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.