Maximal fluctuations around the Wulff shape for edge-isoperimetric sets in ${\mathbb{Z}}^d$: a sharp scaling law

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