From local giants to locality in long-range percolation
We prove the analogue of Schramm's locality conjecture for long-range percolation on transitive graphs of polynomial growth with $α\in (0,2)$. In this setting, we also prove the joint continuity of the percolation probability $θ$ with respect to three parameters: the underlying graph with respect to the local topology, the connectivity kernel, and the percolation parameter $β$ for all values of $β\in \mathbf{R}_+$, including the critical parameter $β_c$. We also prove a number of results related to the supercritical sharpness of long-range percolation: the long-range order decay of the distribution of finite clusters, the truncation problem, the anchored isoperimetric dimension and the transience of the infinite percolation cluster, and the smoothness of the percolation characters. We obtain these results from proving the local existence-and-uniqueness of the linear-sized (giant) cluster. As an immediate corollary of the local existence-and-uniqueness of the giant we obtain the law of large numbers, which answers a special case of a question of Nekrashevych and Pete \cite[Question 1.3]{nekrashevych_scale-invariant_2011}. The main technical contribution is the construction of a renormalisation scheme combining iteratively merged Voronoi tiles with scale-invariant nets, related to the scale-invariant groups of Benjamini.