论文标题
远程渗透的关键指数的等量术的下限
Isoperimetric lower bounds for critical exponents for long-range percolation
论文作者
论文摘要
我们在$ \ mathbb {z}^d $上研究独立的远程渗透，其中$ x $和$ y $与概率$ 1-e^{ - β\ | x-y \ | x-y \ |^{ - d-α}} $连接，用于$ 0> 0 $。提供了由$δ= \ lim_ {n \ to \ infty} \ frac { - \ log（n）} {\ log \ left（\ mathbb {p} _ {β_c} _ {β_c} \ lest（| k_0 | wef）| \ lim_ {x \ to \ infty} \ frac {\ log \ left（\ Mathbb {p} _ {β_c} \ left（0 \ leftrightArrow x \ right）\ right）\ right）}} {\ log log（\ | x \ | x \ | | x \ |）} + d $ restion，n $ k__0 $ \ begin {equation*} δ\ geq \ frac {d+（α\ wedge 1）} {d-（α\ wedge 1）} \ \ text {and} \ 2- \ 2- \ 2- \geQα\ wedge 1 \ text。 \ end {equation*} $δ$上的下限被认为是$ d = 1，α\ in \ weft [\ frac {1} {3} {3}，1 \右）$的，而$ d = 2，α\ in \ in \ weft [\ frac {\ frac {2} {2} {3} {3} {3} {3}，1 \ right] $ ins poss in $ 2-- $ 2- $ d = $ 2- $ d = $ 2- $ （0,1）$，对于$ d> 1 $，对于$α\ in \ weft（0,1 \ right] $，否则不认为否则是鲜明的。我们的主要工具是关键指数与$ \ Mathbb {z}^d $的关键指数与等级之间的连接。
We study independent long-range percolation on $\mathbb{Z}^d$ where the vertices $x$ and $y$ are connected with probability $1-e^{-β\|x-y\|^{-d-α}}$ for $α> 0$. Provided the critical exponents $δ$ and $2-η$ defined by $δ= \lim_{n\to \infty} \frac{-\log(n)}{\log\left(\mathbb{P}_{β_c}\left(|K_0|\geq n\right)\right)}$ and $2-η= \lim_{x \to \infty} \frac{\log\left(\mathbb{P}_{β_c}\left(0\leftrightarrow x\right)\right)}{\log(\|x\|)} + d$ exist, where $K_0$ is the cluster containing the origin, we show that \begin{equation*} δ\geq \frac{d+(α\wedge 1)}{d-(α\wedge 1)} \ \text{ and } \ 2-η\geq α\wedge 1 \text. \end{equation*} The lower bound on $δ$ is believed to be sharp for $d = 1, α\in \left[\frac{1}{3},1\right)$ and for $d = 2, α\in \left[\frac{2}{3},1\right]$, whereas the lower bound on $2-η$ is sharp for $d=1, α\in (0,1)$, and for $α\in \left(0,1\right]$ for $d>1$, and is not believed to be sharp otherwise. Our main tool is a connection between the critical exponents and the isoperimetry of cubes inside $\mathbb{Z}^d$.
