Distances in $\frac{1}{\mathrm{\mid }x-y{\mathrm{\mid }}^{2d}}$ percolation models for all dimensions

We study independent long-range percolation on ${\mathbb{Z}}^d$ for all dimensions $d$, where the vertices $u$ and $v$ are connected with probability 1 for $\mathrm{\parallel }u-v{\mathrm{\parallel }}_{\mathrm{\infty }}=1$ and with probability $p(\beta ,\{u,v\})=1-e^{-\beta {\int }_{u+{[0,1)}^d}{\int }_{v+{[0,1)}^d}\frac{1}{\mathrm{\parallel }x-y{\mathrm{\parallel }}_2^{2d}}dxdy}\approx \frac{\beta }{\mathrm{\parallel }u-v{\mathrm{\parallel }}_2^{2d}}$ for $\mathrm{\parallel }u-v{\mathrm{\parallel }}_{\mathrm{\infty }}\ge 2$. Let $u\in {\mathbb{Z}}^d$ be a point with $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}=n$. We show that both the graph distance $D(\mathbf{0},u)$ between the origin $\mathbf{0}$ and $u$ and the diameter of the box $\{0,\dots ,n{\}}^d$ grow like $n^{\theta (\beta )}$, where $0<\theta (\beta )<1$. We also show that the graph distance and the diameter of boxes have the same asymptotic growth when two vertices $u,v$ with $\mathrm{\parallel }u-v{\mathrm{\parallel }}_2>1$ are connected with a probability that is close enough to $p(\beta ,\{u,v\})$. Furthermore, we determine the asymptotic behavior of $\theta (\beta )$ for large $\beta$, and we discuss the tail behavior of $\frac{D(\mathbf{0},u)}{\mathrm{\parallel }u{\mathrm{\parallel }}_2^{\theta (\beta )}}$.