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

We study independent long-range percolation on Z d for all dimensions d , where the vertices u and v are connected with probability 1 for ‖ u - v ‖ ∞ = 1 and with probability 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 . Let u ∈ Z d be a point with ‖ u ‖ ∞ = n . We show that both the graph distance D ( 0 , u ) between the origin 0 and u and the diameter of the box { 0 , … , n } d grow like n θ ( β ) , where 0 < θ ( β ) < 1 . We also show that the graph distance and the diameter of boxes have the same asymptotic growth when two vertices u , v with ‖ u - v ‖ 2 > 1 are connected with a probability that is close enough to p ( β , { u , v } ) . Furthermore, we determine the asymptotic behavior of θ ( β ) for large β , and we discuss the tail behavior of D ( 0 , u ) ‖ u ‖ 2 θ ( β ) .