Cluster-size decay in supercritical long-range percolation

We study the cluster-size distribution of supercritical long-range percolation on $\mathbb{Z}^d$, where two vertices $x,y\in\mathbb{Z}^d$ are connected by an edge with probability $\mathrm{p}(\|x-y\|):=p\min(1,\beta\|x-y\|)^{-d\alpha}$ for parameters $p\in(0, 1]$, $\alpha>1$, and $\beta>0$. We show that when $\alpha>1+1/d$, and either $\beta$ or $p$ is sufficiently large, the probability that the origin is in a finite cluster of size at least $k$ decays as $\exp\big(-\Theta(k^{(d-1)/d})\big)$. This corresponds to classical results for nearest-neighbor Bernoulli percolation on $\mathbb{Z}^d$, but is in contrast to long-range percolation with $\alpha<1+1/d$, when the exponent of the stretched exponential decay changes to $2-\alpha$. This result, together with our accompanying paper, establishes the phase diagram of long-range percolation with respect to cluster-size decay. Our proofs rely on combinatorial methods that show that large delocalized components are unlikely to occur. As a side result we determine the asymptotic growth of the second-largest connected component when the graph is restricted to a finite box.