Power-law bounds for critical long-range percolation below the upper-critical dimension

Abstract

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 (CMP, 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 \[ \mathbf{P}_{β_c}\bigl(|K|\geq n\bigr) \leq C n^{-(d-α)/(2d+α)} \] 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 $\eta$.

34 pages, 3 figures. Complete proof of a power-law upper bound under the stronger assumption that alpha<d/4 in the first 9 pages. V2: Minor corrections and improvements throughout. Added a lot of detail to Section 3, which is now self-contained. Accepted version, to appear in PTRF