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

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

arXiv comment70 pages, 9 figures

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Cited by in corpus (2)

• Arithmetic oscillations of the chemical distance in long-range percolation on ${\mathbb{Z}}^d$
• Continuity of the critical value and a shape theorem for long-range percolation