论文标题
$ \ frac {1} {| x-y |^{2d}} $ percolation模型的距离
Distances in $\frac{1}{|x-y|^{2d}}$ percolation models for all dimensions
论文作者
论文摘要
我们在$ \ mathbb {z}^d $上研究所有尺寸$ d $的独立远程渗透，其中Vertices $ u $和$ v $与概率1连接，$ \ | u-v \ | _ | _ \ | _ \ infty = 1 $，并且概率是$ p（β，\ {u，v \}）= 1- e^{ - β\ int_ {u++\ left [0,1 \ right）^d}^d} \ int_ {v+\ left [0,1 \ off [0,1 \ oright）^d} \ frac {1} \fracβ{\ | u-v \ | _2^{2d}} $ for $ \ | | u-v \ | _ \ | _ \ infty \ geq 2 $。令$ u \ in \ mathbb {z}^d $是$ \ | u \ | _ \ infty = n $的点。我们表明，在原点$ \ mathbf {0} $和$ u $和盒子$ \ {0，\ ldots，n \}^d $中，$ n \}^d $在$ n^{θ（β）$ 0 <1 $ $ 0 <umy u $的情况下，$ n \}^d $在其中$ n \}^d $之间，$ n \}^d $之间的图形距离$ d（\ mathbf {0}，u），u）$在盒子$ \ {0，\ ldots，n \}^d $之间我们还表明，当两个顶点$ u，带有$ \ | u-v \ | _2> 1 $的两个顶点$ u，v $时，盒子的距离和直径具有相同的渐近生长。此外，我们确定了大$β$的$θ（β）$的渐近行为，并讨论了$ \ frac {d（\ mathbf {0}，u）} {\ | U \ | _2^{θ（β（β）}} $的尾巴行为。
We study independent long-range percolation on $\mathbb{Z}^d$ for all dimensions $d$, where the vertices $u$ and $v$ are connected with probability 1 for $\|u-v\|_\infty=1$ and with probability $p(β,\{u,v\})=1-e^{-β\int_{u+\left[0,1\right)^d} \int_{v+\left[0,1\right)^d} \frac{1}{\|x-y\|_2^{2d}}d x d y } \approx \fracβ{\|u-v\|_2^{2d}}$ for $\|u-v\|_\infty \geq 2$. Let $u \in \mathbb{Z}^d$ be a point with $\|u\|_\infty=n$. We show that both the graph distance $D(\mathbf{0},u)$ between the origin $\mathbf{0}$ and $u$ and the diameter of the box $\{0 ,\ldots, n\}^d$ grow like $n^{θ(β)}$, where $0<θ(β) < 1$. We also show that the graph distance and the diameter of boxes have the same asymptotic growth when two vertices $u,v$ with $\|u-v\|_2 > 1$ are connected with a probability that is close enough to $p(β,\{u,v\})$. Furthermore, we determine the asymptotic behavior of $θ(β)$ for large $β$, and we discuss the tail behavior of $\frac{D(\mathbf{0},u)}{\|u\|_2^{θ(β)}}$.
