Behavior of the distance exponent for $\frac{1}{|x-y|^{2d}}$ long-range percolation

TL;DR

This paper investigates the behavior of the distance exponent in long-range percolation models on integer lattices, showing its continuity, monotonicity, and asymptotic behavior for small parameters.

Contribution

It proves the continuity and strict decrease of the distance exponent with respect to the parameter , and derives its asymptotic form in one dimension for small .

Findings

The distance exponent (d,) is continuous and strictly decreasing in .

In one dimension, (1,) \u2264 1-+o() for small .

The graph distance scales as  ext{distance}^{(d,)}.

Abstract

We study independent long-range percolation on $\mathbb{Z}^d$ where the vertices $u$ and $v$ are connected with probability asymptotic to $\frac{\beta}{\|u-v\|^{2d}}$ for $\|u-v\|_\infty\geq 2$ and with probability 1 for $\|u-v\|_\infty=1$, where $\beta \geq 0$ is a parameter. It is proven in [5] that there exists an exponent $\theta=\theta(d,\beta) \in \left(0,1\right]$ such that the graph distance between the origin $\mathbf{0}$ and $x \in \mathbb{Z}^d$ scales like $\|x\|^{\theta}$. We prove that this exponent $\theta(d,\beta)$ is continuous and strictly decreasing as a function in $\beta$. Furthermore, we show that $\theta(d,\beta)=1-\beta+o(\beta)$ for small $\beta$ in dimension $d=1$.