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

TL;DR

This paper investigates long-range percolation on integer lattices across all dimensions, revealing how graph distances and diameters grow polynomially with distance, and analyzing the asymptotic behavior of these growth rates.

Contribution

It provides a comprehensive analysis of the growth of graph distances and diameters in long-range percolation models for all dimensions, including asymptotics of the growth exponent.

Findings

Graph distance and diameter grow like n^{θ(β)} with 0<θ(β)<1.

Graph distances and diameters have the same asymptotic growth under certain connection probabilities.

Asymptotic behavior of θ(β) is characterized for large β.

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 $\frac{D(\mathbf{0},u)}{\|u\|_2^{\theta(\beta)}}$.