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 1 for $|u-v|\infty=1$ and with probability $1-e^{-\beta \int{u+\left[0,1\right)^d} \int_{u+\left[0,1\right)^d} \frac{1}{|x-y|^{2d}} d x d y } \approx \frac{\beta}{|u-v|^{2d}}$ for $|u-v|_\infty\geq 2$, where $\beta \geq 0$ is a parameter. There exists an exponent $\theta=\theta(\beta) \in \left(0,1\right]$ such that the graph distance between the origin $\mathbf{0}$ and $v \in \mathbb{Z}^d$ scales like $|v|^{\theta}$. We prove that this exponent $\theta(\beta)$ is continuous and strictly decreasing as a function in $\beta$. Furthermore, we show that $\theta(\beta)=1-\beta+o(\beta)$ for small $\beta$ in dimension $d=1$.