从局部巨集到长程渗流中的局部性
From local giants to locality in long-range percolation
AI总结：
研究在特定多项式增长传递图上长程渗流的Schramm局部性猜想类似物，通过构建重整化方案证明相关结果，包括渗流概率联合连续性、超临界锐度等，还得出大数定律，回答了特定问题。
AI中文摘要：
我们证明了在具有\(\alpha \in (0,2)\)的多项式增长传递图上长程渗流的Schramm局部性猜想的类似物。在此设定下，我们还证明了渗流概率\(\theta\)关于三个参数的联合连续性：相对于局部拓扑的基础图、连通性核以及对于所有\(\beta \in \mathbf{R}_+\)值（包括临界参数\(\beta_c\)）的渗流参数\(\beta\)。我们还证明了一些与长程渗流的超临界锐度相关的结果：有限簇分布的长程序衰减、截断问题、锚定等周维数和无限渗流簇的暂态性以及渗流特征的光滑性。我们通过证明线性大小（巨集）簇的局部存在性和唯一性获得这些结果。作为巨集局部存在性和唯一性的直接推论，我们得到了大数定律，这回答了Nekrashevych和Pete \cite[问题1.3]{nekrashevych_scale - invariant_2011}的一个特殊情况。主要技术贡献是构建了一种重整化方案，该方案将迭代合并的Voronoi瓷砖与与Benjamini的尺度不变群相关的尺度不变网相结合。
英文摘要：
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.
