Freezing Phase Transitions for Lattice Systems and Higher-Dimensional Subshifts

Let $X = \mathcal{A}^{\mathbb{Z}^d}$, where $d \geq 1$ and $\mathcal{A}$ is a finite set, equipped with the action of the shift map. For a given continuous potential $\phi: \mathcal{A}^{\mathbb{Z}^d} \to \mathbb{R}$ and $\beta>0$ (``inverse temperature''), there exists a (nonempty) set of equilibrium states $\mathrm{ES}(\beta\phi)$. The potential $\phi$ is said to exhibit a ``freezing phase transition'' if $\mathrm{ES}(\beta\phi) = \mathrm{ES}(\beta'\phi)$ for all $\beta, \beta' > \beta_c$, while $\mathrm{ES}(\beta\phi) \neq \mathrm{ES}(\beta'\phi)$ for any $\beta < \beta_c < \beta'$, where $\beta_c\in (0,\infty)$ is a critical inverse temperature depending on $\phi$. In this paper, given any proper subshift $X_0$ of $X$, we explicitly construct a continuous potential $\phi: X \to \mathbb{R}$ for which there exists $\beta_c \in (0,\infty)$ such that $\mathrm{ES}(\beta\phi)$ coincides with the set of measures of maximal entropy on $X_0$ for all $\beta > \beta_c$, whereas for all $\beta < \beta_c$, $\mu(X_0)=0$ for all $\mu\in \mathrm{ES}(\beta\phi)$. This phenomenon was previously studied only for $d = 1$ in the context of dynamical systems and for restricted classes of subshifts, with significant motivation stemming from quasicrystal models. Additionally, we prove that under a natural summability condition -- satisfied, for instance, by finite-range potentials or exponentially decaying potentials -- freezing phase transitions are impossible.