Distances in 1 x-y ²ᵈ Percolation Models for all Dimensions

We study independent long-range percolation on Zᵈ Z d for all dimensions d , where the vertices u and v are connected with probability 1 for u-v _∞ =1 ‖ u - v ‖ ∞ = 1 and with probability p(β ,,v\)=1-e^-β ∫ u+[ 0,1) ᵈ ∫ v+[ 0,1) ᵈ 1 x-y ₂²ᵈd x d y ≈ β u-v ₂²ᵈ p ( β , { u , v } ) = 1 - e - β ∫ u + 0 , 1 d ∫ v + 0 , 1 d 1 ‖ x - y ‖ 2 2 d d x d y ≈ β ‖ u - v ‖ 2 2 d for u-v _∞ ≥ 2 ‖ u - v ‖ ∞ ≥ 2 . Let u ∈ Zᵈ u ∈ Z d be a point with u _∞ =n ‖ u ‖ ∞ = n . We show that both the graph distance D(0,u) D ( 0 , u ) between the origin 0 0 and u and the diameter of the box \0,… , n\ᵈ { 0 , … , n } d grow like nθ (β ) n θ ( β ) , where 0<θ (β ) < 1 <mml:math xmlns: