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 $\frac{\beta }{\Vert u-v{\Vert }^{2d}}$ for $\Vert u-v{\Vert }_{\infty }\ge 2$ and with probability 1 for $\Vert u-v{\Vert }_{\infty }=1$, where $\beta \ge 0$ 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 $0$ and $x\in {\mathbb{Z}}^d$ scales like $\Vert x{\Vert }^{\theta }$. 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 $d=1$.

arXiv comment67 pages. Accepted in Electronic Journal of Probability

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