Freezing Phase Transitions for Lattice Systems and Higher-Dimensional Subshifts
Bibliographic record
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.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".