论文标题
关键的远程渗透的幂律界限以下
Power-law bounds for critical long-range percolation below the upper-critical dimension
论文作者
论文摘要
我们在$ \ mathbb {z}^d $上研究远程bernoulli渗透，其中每个两个顶点$ x $和$ y $都由带有概率$ 1- \ exp（-β\ | x-y \ |^|^{ - d-α}）的边缘连接。 Noam Berger（CMP，2002）的定理是，如果$ 0 <α<d $，那么关键参数$β_C$上没有无限群集。我们给出了该定理的新的定量证明，以建立幂律上限\ [\ mathbf {p} _ {β_c} \ bigl（| k | | | | \ geq n \ bigr）\ leq c n^{ - （d-d-α）/（dd+α）/（2d+α）/（2d+α）/（2d+α）}} $ n \ geq的$ kq $ kq $ kq 1 $ kq。我们认为，这是Bernoulli渗透模型的第一个严格的幂律上限，它既不是平面也不期望表现出均值场临界行为。 作为证明的一部分，我们建立了通用不平等，这意味着在任何有限图上渗透的最大群集的最大大小与具有高概率的平均值相同。我们应用这种不等式来得出新的严格的超标准不等式$（2-η）（δ+1）\ leq d（δ-1）$，与群集数量指数$δ$和两点函数指数$η$相关。
We study long-range Bernoulli percolation on $\mathbb{Z}^d$ in which each two vertices $x$ and $y$ are connected by an edge with probability $1-\exp(-β\|x-y\|^{-d-α})$. It is a theorem of Noam Berger (CMP, 2002) that if $0<α<d$ then there is no infinite cluster at the critical parameter $β_c$. We give a new, quantitative proof of this theorem establishing the power-law upper bound \[ \mathbf{P}_{β_c}\bigl(|K|\geq n\bigr) \leq C n^{-(d-α)/(2d+α)} \] for every $n\geq 1$, where $K$ is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $(2-η)(δ+1)\leq d(δ-1)$ relating the cluster-volume exponent $δ$ and two-point function exponent $η$.
