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

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 $d=1$.