Gravitational-field variational-shape conjecture for the Ising phase separation

Let $g_N=g/N^2$ with $g<0$, and let $h_N$ be the magnetic field specified in the source. For $\beta>\beta_{\mathrm{c}}$, let $\beta_{\mathrm{c}}$ be the critical inverse temperature, let $\mu^-_{N,\beta,h_N,m}$ be the relevant Ising measure with minus boundary condition and magnetization parameter $m$, let $\Gamma_N$ be the relevant interface or phase-separation set, let $\mathsf{m}_N(x)$ be the associated microscopic magnetization profile, and let $q_{V_*}(x)$ be the profile associated with a minimizer $V_*$. Then, for every $\epsilon>0$,

$$\lim_{N\to\infty} \mu^-_{N,\beta,h_N,m}\Bigl( \inf_{V_*} \sum_{x\in\Gamma_N} \bigl\lvert \mathsf{m}_N(x) - q_{V_*}(x) \bigr\rvert \leq \epsilon \lvert \Gamma_N \rvert \Bigr) = 1,$$

where the infimum is over all minimizers $V_*$ of

$$\text{minimize }\int_{\partial V} \tau_\beta(\vec n_s)\,\mathrm{d}\mathcal{H}^{(d-1)}_s + 2m_\beta^*g\int_V x_d\,\mathrm{d}x$$

over all subsets $V\subset[0,1]^d$ satisfying

$$\mathcal{H}^{(d)}(V)=\frac{m_\beta^*+m}{2m_\beta^*}.$$

Gravitational-field variational-shape conjecture. The preceding convergence in probability holds. The supplied context says that the solution of this variational problem is known in dimension $d=2$ when the constraint $V\subset[0,1]^2$ is dropped, or when $m$ is sufficiently close to $-m_\beta^*$ for that solution to fit inside the box; the conjecture concerns the corresponding phase-separated geometry in the general setting.