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

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Russo's formula for expectations
• Lemma 3.1
• Remark 3.2
• proof : Proof of Theorem \ref{['theo:smallbeta']}
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3