Maximal Fluctuations Around the Wulff Shape for Edge-Isoperimetric Sets in $$\varvec{{\mathbb {Z}}^d}$$: A Sharp Scaling Law

We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice $${\mathbb {Z}}^d$$ 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 $$O(n^{(d-1+2^{1-d})/d})$$ O ( n ( d - 1 + 2 1 - d ) / d ) lattice points and that the exponent $$(d-1+2^{1-d})/d$$ ( d - 1 + 2 1 - d ) / d is optimal. This extends the previously found ‘$$n^{3/4}$$ n 3 / 4 laws’ for $$d=2,3$$ 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.