${\mathbb Z^d}$中边等周集在Wulff形附近的最大涨落：一个尖锐的标度律
Maximal fluctuations around the Wulff shape for edge-isoperimetric sets in ${\mathbb Z^d}$: a sharp scaling law
摘要：
我们推导出一个尖锐的标度律的偏差的边缘等周集的格子$\mathbb Z^d$从限制武尔夫形状在任意尺寸。当元素个数$n$发散时，我们证明了对应的Wulff集的对称差至多由$O(n^{(d-1+2^{1-d})/d})$格点组成，并且指数$(d-1+2^{1-d})/d$是最优的.这将以前发现的$d=2,3$的“$n^{3/4}$定律”扩展到一般维度。因此，我们得到最佳估计的收敛速度限制伍尔夫形状的$n$发散。
英文摘要：
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.
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