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

We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice Zd 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+21−d )/d ) lattice points and that the exponent (d − 1 + 21−d )/d is optimal. This extends the previously found ‘n3/4 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.