*Published Paper*

**Inserted:** 4 mar 2020

**Last Updated:** 1 dec 2020

**Journal:** Comm. Math. Phys.

**Volume:** 380

**Pages:** 947–971

**Year:** 2020

**Doi:** 10.1007/s00220-020-03879-x

**Abstract:**

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 $O(n(d−1+21−d)/d)$ lattice points and that the exponent $(d−1+21−d)/d$ 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.