Freezing Phase Transitions for Lattice Systems and Higher-Dimensional Subshifts

Abstract

Let $X={\mathcal{A}}^{{\mathbb{Z}}^d}$, where $d\ge 1$ and $\mathcal{A}$ is a finite set, equipped with the action of the shift map. For a given continuous potential $\varphi :{\mathcal{A}}^{{\mathbb{Z}}^d}\to \mathbb{R}$ and $\beta >0$ (``inverse temperature''), there exists a (nonempty) set of equilibrium states $\mathrm{ES}(\beta \varphi )$. The potential $\varphi$ is said to exhibit a ``freezing phase transition'' if $\mathrm{ES}(\beta \varphi )=\mathrm{ES}({\beta }^{\prime }\varphi )$ for all $\beta ,{\beta }^{\prime }>{\beta }_c$, while $\mathrm{ES}(\beta \varphi )\ne \mathrm{ES}({\beta }^{\prime }\varphi )$ for any $\beta <{\beta }_c<{\beta }^{\prime }$, where ${\beta }_c\in (0,\infty )$ is a critical inverse temperature depending on $\varphi$. In this paper, given any proper subshift $X_0$ of $X$, we explicitly construct a continuous potential $\varphi :X\to \mathbb{R}$ for which there exists ${\beta }_c\in (0,\infty )$ such that $\mathrm{ES}(\beta \varphi )$ 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 $\mu \in \mathrm{ES}(\beta \varphi )$. 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.

43 pages, 2 figures