AbstractWe study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$
Z
d
in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$
1
-
exp
(
-
β
‖
x
-
y
‖
-
d
-
α
)
. It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$
0
<
α
<
d
then there is no infinite cluster at the critical parameter $$\beta _c$$
β
c
. We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$
P
β
c
(
|
K
|
≥
n
)
≤
C
n
-
(
d
-
α
)
/
(
2
d
+
α
)
for every $$n\ge 1$$
n
≥
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 $$(2-\eta )(\delta +1)\le d(\delta -1)$$
(
2
-
η
)
(
δ
+
1
)
≤
d
(
δ
-
1
)
relating the cluster-volume exponent $$\delta $$
δ
and two-point function exponent $$\eta $$
η
.
Power-law bounds for critical long-range percolation below the upper-critical dimension
