论文标题
$ \ frac {1} {| x-y |^{2d}} $远程渗透的距离指数的行为
Behavior of the distance exponent for $\frac{1}{|x-y|^{2d}}$ long-range percolation
论文作者
论文摘要
我们在$ \ mathbb {z}^d $上研究独立的远程渗透，其中顶点$ u $和$ v $与概率1相连，$ \ | u-v \ | __ \ | _ \ iffty = 1 $，并且概率$ 1-e^{ - β\β\ int_ {uint_ {u+\ weft [u+\ weft [u+\ weft [u+\ weft [u+\ weft [u+\ weft [u+\ right]^d}^d} \ int_ {u+\ weft [0,1 \ right）^d} \ frac {1} {\ | x-y \ |^{2d}} d x d y} \近似\fracβ{\ | 0 $是一个参数。存在一个指数$θ=θ（β）\ in \左（0,1 \右）$，使得原点$ \ m athbf {0} $和$ v \ in \ mathbb {z}^d $ scales scale $ c \ | v \ | v \ |^θ$ scrip of thit deccons $ unigtion and contery and contery和contern content and contery和content contery和contention $ unifucty and contery和$β$。
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 $1-e^{-β\int_{u+\left[0,1\right)^d} \int_{u+\left[0,1\right)^d} \frac{1}{\|x-y\|^{2d}} d x d y } \approx \fracβ{\|u-v\|^{2d}}$ for $\|u-v\|_\infty\geq 2$, where $β\geq 0$ is a parameter. There exists an exponent $θ=θ(β) \in \left(0,1\right]$ such that the graph distance between the origin $\mathbf{0}$ and $v \in \mathbb{Z}^d$ scales like $\|v\|^θ$. We prove that this exponent $θ(β)$ is continuous and strictly decreasing as a function in $β$. Furthermore, we show that $θ(β)=1-β+o(β)$ for small $β$ in dimension $d=1$.
