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 \(1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })\). It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if \(0<\alpha <d\) then there is no infinite cluster at the critical parameter \(\beta _c\). We give a new, quantitative proof of this theorem establishing the power-law upper bound

for every \(n\ge 1\), 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 \((2-\eta )(\delta +1)\le d(\delta -1)\) relating the cluster-volume exponent \(\delta \) and two-point function exponent \(\eta \).

我们研究\({\mathbb {Z}}^d\)上的长程伯努利渗流，其中每两个顶点x和y由一条边以概率\(1-\exp (-\beta \Vert xy\垂直 ^{-d-\alpha })\)。Noam Berger ( Commun. Math. Phys. , 2002) 的一个定理是，如果\(0<\alpha <d\)那么在临界参数\(\beta _c\)处不存在无限集群。我们对建立幂律上限的定理给出了一个新的定量证明

对于每个\(n\ge 1\)，其中K是原点的簇。我们相信这是伯努利渗透模型的第一个严格的幂律上限，该模型既不是平面的，也不是预期表现出平均场临界行为的。作为证明的一部分，我们建立了一个普遍的不等式，这意味着在任何有限图上渗透的集群的最大大小与它的平均值具有相同的顺序，概率很高。我们应用这个不等式推导出一个新的严格的超标度不等式\((2-\eta )(\delta +1)\le d(\delta -1)\)将簇体积指数\(\delta \)和两个-point 函数指数\(\eta \)。