Distances in $$\frac{1}{\Vert x-y\Vert ^{2d}}$$ Percolation Models for all Dimensions

Abstract

AbstractWe study independent long-range percolation on $$\mathbb {Z}^d$$ Z d for all dimensions d, where the vertices u and v are connected with probability 1 for $$\Vert u-v\Vert _\infty =1$$ ‖ u - v ‖ ∞ = 1 and with probability $$p(\beta ,\{u,v\})=1-e^{-\beta \int _{u+\left[ 0,1\right) ^d} \int _{v+\left[ 0,1\right) ^d} \frac{1}{\Vert x-y\Vert _2^{2d}}d x d y } \approx \frac{\beta }{\Vert u-v\Vert _2^{2d}}$$ 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 $$\Vert u-v\Vert _\infty \ge 2$$ ‖ u - v ‖ ∞ ≥ 2 . Let $$u \in \mathbb {Z}^d$$ u ∈ Z d be a point with $$\Vert u\Vert _\infty =n$$ ‖ u ‖ ∞ = n . We show that both the graph distance $$D(\textbf{0},u)$$ D ( 0 , u ) between the origin $$\textbf{0}$$ 0 and u and the diameter of the box $$\{0,\ldots , n\}^d$$ { 0 , … , n } d grow like $$n^{\theta (\beta )}$$ n θ ( β ) , where $$0<\theta (\beta ) < 1$$ 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 $$\Vert u-v\Vert _2 > 1$$ ‖ u - v ‖ 2 > 1 are connected with a probability that is close enough to $$p(\beta ,\{u,v\})$$ p ( β , { u , v } ) . Furthermore, we determine the asymptotic behavior of $$\theta (\beta )$$ θ ( β ) for large $$\beta $$ β , and we discuss the tail behavior of $$\frac{D(\textbf{0},u)}{\Vert u\Vert _2^{\theta (\beta )}}$$ D ( 0 , u ) ‖ u ‖ 2 θ ( β ) .