Maximal Fluctuations Around the Wulff Shape for Edge-Isoperimetric Sets in $$\varvec{{\mathbb {Z}}^d}$$: A Sharp Scaling Law

Abstract

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

Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy

“…We also mention extensions to other crystals [16,40,42] and dimers [26,27]. By way of contrast, in dimension three or higher the recent results [11,39,41] characterize optimal energy configurations within classes of lattices and are in this sense conditional to crystallization.…”

The double-bubble problem on the square lattice

“…The presence of interspecific interactions adds some level of rigidity. This is revealed by the fact, which we prove, that the distance between different ground states scales like N 1/2 , in contrast with the purely edge-isoperimetric case, where fluctuations are of order N 3/4 [24], see also [9,13,25,31].…”

Microscopical Justification of the Winterbottom problem for well-separated Lattices

“…where I n,Λ,x F is defined by (22), Γ-converges with respect to the weak* convergence of measures to the functional I ∞,i defined by…”

“…For the justification of the problem in dimension d = 2 in the context of statistical mechanics and the Ising model we refer to the review [9] (see also [14,17]) for the Wulff shape in the scaling limit at low-temperature and to [4,23,24] for the setting related to the Winterbottom shape, while in the context of atomistic models a rigorous discrete to continuum passage for triangular reference lattices has been first carried out by means of Γ-convergence in [2] for the Wulff shape and then extended to the Winterbottom situation in [25]. The emergence of the Wulff shape has been also deduced for the square lattice in [20,21] and the hexagonal lattice [7] by means of a different approach based on induction techniques related to the crystallization problem [13], and of the quantification of the deviation of discrete ground states from the asymptotic Wulff shape in the so called n 3/4 law (where n is the number of atoms), which was previously introduced in [26], and then extended to those settings (see also [8]), and more recently to higher dimensions in [19,22]. A derivation by Γ-convergence of an energy of the type (1) coupled with a bulk elastic term in the context of models for epitaxially-strained thin films introduced in [5,6,11,27,28] has been instead determined in [18] under a graph constraint for the region occupied by the film drop.…”