Distances in $\frac{1}{|x-y|^{2d}}$ percolation models for all dimensions (2208.04800v3)

Abstract: 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 $|u-v|\infty=1$ and with probability $p(\beta,{u,v})=1-e^{-\beta \int{u+\left[0,1\right)^d} \int_{v+\left[0,1\right)^d} \frac{1}{|x-y|2^{2d}}d x d y } \approx \frac{\beta}{|u-v|_2^{2d}}$ for $|u-v|\infty \geq 2$. Let $u \in \mathbb{Z}^d$ be a point with $|u|_\infty=n$. We show that both the graph distance $D(\mathbf{0},u)$ between the origin $\mathbf{0}$ and $u$ and the diameter of the box ${0 ,\ldots, n}^d$ 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 $|u-v|_2 > 1$ 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 $\frac{D(\mathbf{0},u)}{|u|_2^{\theta(\beta)}}$.