论文标题
层次渗透中的关键集群体积
Critical cluster volumes in hierarchical percolation
论文作者
论文摘要
我们考虑在$ d $二维的层次晶格上的远距离bernoulli债券渗透，其中每对点$ x $和$ y $由Edge连接，概率$ 1- \ exp（-β\ | x-y \ | x-y \ |^|^|^{ - d-α}）我们在该模型的临界点$β=β_C$中研究簇的体积，证明了对盒子内原点群集群体积的所有顺序的精确估计。我们应用这些估计值以证明对原始簇的尾部的最新估计，表示为$ k $，以关键性，即\ [\ mathbb {p} _ {β_c} _ {β_c}（| k | \ geq N） n^{ - 1/2}（\ log n）^{1/4}＆d =3α\\ n^{ - 1/2}＆d>3α。 \ end {case} \尤其是，当$ d $低于限制性d_c =3α$时，我们计算为$（d+α）/（d-α）/（d-α）$的关键指数$δ$，并确定在上层临界自身处缩放到缩放量表的精确顺序。有趣的是，我们发现，这些多毛的校正并不是预计将对$ \ mathbb {z}^6 $ by Essam，gaunt和Guttmann（J.Phys。A1978）的$ \ mathbb {z}^6 $持有的那些校正。我们的工作还为研究模型的缩放限制的基础奠定了基础：在高维情况下，我们证明，我们证明，在适当的正常化下，盒子内部群中的大小偏见的分布分布到chi-squared随机变量下，而在低维情况下，我们在列出$ d <3α$中均构成了一个适当的范围，以构成一定的范围。 $ \ ell^p \ setMinus \ {0 \} $，仅当$ p> 2d/（d+α）$。
We consider long-range Bernoulli bond percolation on the $d$-dimensional hierarchical lattice in which each pair of points $x$ and $y$ are connected by an edge with probability $1-\exp(-β\|x-y\|^{-d-α})$, where $0<α<d$ is fixed and $β\geq 0$ is a parameter. We study the volume of clusters in this model at its critical point $β=β_c$, proving precise estimates on the moments of all orders of the volume of the cluster of the origin inside a box. We apply these estimates to prove up-to-constants estimates on the tail of the volume of the cluster of the origin, denoted $K$, at criticality, namely \[ \mathbb{P}_{β_c}(|K|\geq n) \asymp \begin{cases} n^{-(d-α)/(d+α)} & d < 3α\\ n^{-1/2}(\log n)^{1/4} & d=3α\\ n^{-1/2} & d>3α. \end{cases} \] In particular, we compute the critical exponent $δ$ to be $(d+α)/(d-α)$ when $d$ is below the upper-critical dimension $d_c=3α$ and establish the precise order of polylogarithmic corrections to scaling at the upper-critical dimension itself. Interestingly, we find that these polylogarithmic corrections are not those predicted to hold for nearest-neighbour percolation on $\mathbb{Z}^6$ by Essam, Gaunt, and Guttmann (J. Phys. A 1978). Our work also lays the foundations for the study of the scaling limit of the model: In the high-dimensional case $d \geq 3α$ we prove that the sized-biased distribution of the volume of the cluster of the origin inside a box converges under suitable normalization to a chi-squared random variable, while in the low-dimensional case $d<3α$ we prove that the suitably normalized decreasing list of cluster sizes in a box is tight in $\ell^p\setminus \{0\}$ if and only if $p>2d/(d+α)$.
