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We derive a sharp scaling law for deviations of edge-isoperimetric sets inthe lattice $\mathbb Z^d$ from the limiting Wulff shape in arbitrarydimensions. As the number $n$ of elements diverges, we prove that the symmetricdifference 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 $d=2,3$ togeneral dimensions. As a consequence we obtain optimal estimates on the rate ofconvergence to the limiting Wulff shape as $n$ diverges.

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