Conjecture on eventual crystalline growth and unique minimizing movement

Let $\varphi$ be an anisotropy and let $\alpha$ be a suitable parameter. For the discrete solutions $\{E^k\}$ of scheme

, write $\mathbb{Z}_e^2$ and $\mathbb{Z}_o^2$ for the even and odd lattice points, $Z(E)$ for the lattice representation of a set $E$, and $d^\varphi$ for the associated anisotropic distance. Let $\mathcal{N}_\alpha^\varphi$ be the even nucleus, and let $P_\alpha^\varphi$ and $v_\alpha^\varphi$ be as in Theorem

. Conjecture on crystalline growth. Under suitable assumptions on $\varphi$ and for suitable values of $\alpha$, the discrete solutions satisfy either

$$Z(E_\alpha^k)=\left\{\mathbf{i}\in\mathbb{Z}_e^2:d^\varphi(\mathbf{i},E_\alpha^{k-1})<\frac{4}{\alpha}\right\}$$

or

$$Z(E_\alpha^k)=\left\{\mathbf{i}\in\mathbb{Z}_o^2:d^\varphi(\mathbf{i},E_\alpha^{k-1})<\frac{4}{\alpha}\right\}.$$

Moreover, there exists $k_\alpha\in\mathbb{N}$ such that

$$Z(E_\alpha^k)=Z(E_\alpha^{k_\alpha})+\underbrace{\mathcal{N}_\alpha^\varphi+\dots+\mathcal{N}_\alpha^\varphi}_{(k-k_\alpha)\text{-times}},\qquad k\geq k_\alpha.$$

Finally, the limit evolution is the unique minimizing movement $E:[0,+\infty)\to\mathcal{X}$ defined by $E(t)=v_\alpha^\varphi tP_\alpha^\varphi$ for every $t\geq0$. The conjecture predicts eventual parity stabilization and self-similar crystalline growth, together with a uniquely determined limiting motion; the supplied text does not provide evidence resolving it.