Freezing Phase Transitions for Lattice Systems and Higher-Dimensional Subshifts
J. -R. Chazottes, T. Kucherenko, A. Quas
TL;DR
This work establishes that freezing phase transitions can be engineered for lattice systems in higher dimensions by constructing explicit continuous potentials that concentrate equilibrium states onto a prescribed subshift $X_0$ for high inverse temperature $β$ while eliminating the subshift from support below a critical $β_c$. The authors prove a complementary no‑go result: a natural summability condition on the variation of the potential (and on interactions) forbids freezing, ensuring equilibrium states remain fully supported. Their approach combines a constructive dyadic tiling framework with entropy estimates to control where maximizing measures live and to relate freezing to affine segments of the pressure function. The findings illuminate the interplay between dynamics, entropy, and phase structure in multidimensional symbolic systems and connect to quasicrystal models, offering explicit 1D and higher‑dimensional examples and clarifying limitations for finite‑range interactions.
Abstract
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 $φ: \mathcal{A}^{\mathbb{Z}^d} \to \mathbb{R}$ and $β>0$ (``inverse temperature''), there exists a (nonempty) set of equilibrium states $\mathrm{ES}(βφ)$. The potential $φ$ is said to exhibit a ``freezing phase transition'' if $\mathrm{ES}(βφ) = \mathrm{ES}(β'φ)$ for all $β, β' > β_c$, while $\mathrm{ES}(βφ) \neq \mathrm{ES}(β'φ)$ for any $β< β_c < β'$, where $β_c\in (0,\infty)$ is a critical inverse temperature depending on $φ$. In this paper, given any proper subshift $X_0$ of $X$, we explicitly construct a continuous potential $φ: X \to \mathbb{R}$ for which there exists $β_c \in (0,\infty)$ such that $\mathrm{ES}(βφ)$ coincides with the set of measures of maximal entropy on $X_0$ for all $β> β_c$, whereas for all $β< β_c$, $μ(X_0)=0$ for all $μ\in \mathrm{ES}(βφ)$. 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.