Calculus of Variations and Geometric Measure Theory

E. Mainini - B. Schmidt

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

created by mainini on 04 Mar 2020
modified on 01 Dec 2020


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

ArXiv: 2003.01679 PDF


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.