Freezing Phase Transitions for Lattice Systems and Higher-Dimensional Subshifts
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2025
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| author | Chazottes, J. -R. Kucherenko, T. Quas, A. |
| author_facet | Chazottes, J. -R. Kucherenko, T. Quas, A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_20881 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Freezing Phase Transitions for Lattice Systems and Higher-Dimensional Subshifts Chazottes, J. -R. Kucherenko, T. Quas, A. Dynamical Systems Mathematical Physics Probability 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. |
| title | Freezing Phase Transitions for Lattice Systems and Higher-Dimensional Subshifts |
| topic | Dynamical Systems Mathematical Physics Probability |
| url | https://arxiv.org/abs/2504.20881 |