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

We study independent long-range percolation on ${\mathbb{Z}}^d$ where the vertices $u$ and $v$ are connected with probability asymptotic to $\frac{\beta }{\mathrm{\parallel }u-v{\mathrm{\parallel }}^{2d}}$ for $\mathrm{\parallel }u-v{\mathrm{\parallel }}_{\mathrm{\infty }}\ge 2$ and with probability 1 for $\mathrm{\parallel }u-v{\mathrm{\parallel }}_{\mathrm{\infty }}=1$, where $\beta \ge 0$ is a parameter. It is proven in [5] that there exists an exponent $\theta =\theta (d,\beta )\in (0,1]$ such that the graph distance between the origin $\mathbf{0}$ and $x\in {\mathbb{Z}}^d$ scales like $\mathrm{\parallel }x{\mathrm{\parallel }}^{\theta }$. 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 $d=1$.