论文标题
$ {\ mathbb z^d} $围绕wulff形状的最大波动
Maximal fluctuations around the Wulff shape for edge-isoperimetric sets in ${\mathbb Z^d}$: a sharp scaling law
论文作者
论文摘要
我们为晶格$ \ mathbb z^d $在限制性wulff形状中以任意维度的偏差$ \ mathbb z^d $的偏差得出了尖锐的缩放定律。随着元素的数字$ n $分歧，我们证明与相应的WULFF集的对称差异最多由$ o（n^{（d-1+2^{1-d}）/d}）$ lattice点，指数$（d-1+2^{1-d}）/d $是最佳的。这将$ d = 2,3 $的先前发现的“ $ n^{3/4} $ LAWS”扩展到一般尺寸。结果，我们获得了对限制wulff形状的收敛速度的最佳估计，因为$ n $ diverges。
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.
