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arXiv · 2003.01679

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

Edoardo Mainini·Bernd Schmidt
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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 $O(n^{(d-1+2^{1-d})/d})$ lattice points and that the exponent $(d-1+2^{1-d})/d$ is optimal. This extends the previously found `$n^{3/4}$ laws' for $d=2,3$ to general dimensions. As a consequence we obtain optimal estimates on the rate of convergence to the limiting Wulff shape as $n$ diverges.

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Edoardo Mainini, Bernd Schmidt. 2020-03-03. Maximal fluctuations around the Wulff shape for edge-isoperimetric sets in ${\mathbb Z^d}$: a sharp scaling law. https://doi.org/10.1007/s00220-020-03879-x

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