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

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 $η$.