Faceted-droplet hypothesis for the low-temperature three-dimensional Ising model

Let $T$ be sufficiently low. For a configuration $\sigma$ in the constrained periodic ensemble with probability measure $mu_{\Lambda(l),\operatorname{per},T,0}^{<\rho}$, let $\Gamma(\sigma)$ denote the unique macroscopic droplet and let $\lambda K_{\tau}^{<}$ be the corresponding rescaled Wulff shape. The notation $\operatorname{diam}$ denotes diameter and $|\cdot|$ denotes planar area. There are constants $C_1(T)$ and $C_2(T)$ as in the assertion below.

Faceted-droplet hypothesis. As $l\to\infty$, with probability tending to $1$, there exist six distinct two-dimensional planes $L_i=L_i(\sigma)\subset\mathbb{T}^3$, $i=1,\ldots,6$, two perpendicular to each coordinate direction, such that $L_i\cap\Gamma(\sigma)$ are flat facets and, for every $i$,

$$\operatorname{diam}(L_i\cap\Gamma(\sigma))\ge C_1(T)\operatorname{diam}(\lambda K_{\tau}^{<})l,$$

with $C_1(T)\to\sqrt{2/3}$ as $T\to0$;

$$\frac{|L_i\cap\Gamma(\sigma)|}{[\operatorname{diam}(L_i\cap\Gamma(\sigma))]^2}\ge C_2(T),$$

with $C_2(T)\to1/2$ as $T\to0$; and the asymptotic shape of each facet $L_i\cap\Gamma(\sigma)$ is given by the corresponding Wulff construction.

The hypothesis asserts that the macroscopic droplet itself has six coordinate-direction facets, rather than merely resembling the Wulff shape in a weaker sense. It is presented as an expected low-temperature property; the supplied text gives no resolution, so its status remains open.