Freezing Phase Transitions for Lattice Systems and Higher-Dimensional Subshifts
Jean-René Chazottes, Tamara Kucherenko, Anthony Quas · arXiv (Cornell University) · 2025
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 $\mathrm{ES}(βϕ) eq \mathrm{ES}(β'ϕ)$ for any $β β_c$, 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.