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 machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 source (direct Gemma or distilled Codex), 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".