We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice $${\mathbb {Z}}^d$$ from the limiting Wulff shape in arbitrary dimensions. As the number n of elements diverges, we prove that the symmetric difference to the corresponding Wulff set consists of at most $$O(n^{(d-1+2^{1-d})/d})$$ lattice points and that the exponent $$(d-1+2^{1-d})/d$$ is optimal. This extends the previously found ‘ $$n^{3/4}$$ laws’ for $$d=2,3$$ to general dimensions. As a consequence we obtain optimal estimates on the rate of convergence to the limiting Wulff shape as n diverges.

中文翻译：

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$$\varvec{{\mathbb {Z}}^d}$$ 中边等周集的 Wulff 形状周围的最大波动：一个尖锐的标度定律

我们为晶格 $${\mathbb {Z}}^d$$ 中边等周集的偏差从任意维度的限制 Wulff 形状导出了一个尖锐的缩放定律。随着元素数 n 的发散，我们证明了对应的 Wulff 集的对称差异至多由 $$O(n^{(d-1+2^{1-d})/d})$$ 格子组成点并且指数 $$(d-1+2^{1-d})/d$$ 是最优的。这将先前发现的 $$d=2,3$$ 的“$$n^{3/4}$$ 法则”扩展到一般维度。因此，当 n 发散时，我们获得了对限制 Wulff 形状的收敛速度的最佳估计。