Fusion hierarchies, T-systems and Y-systems for the dilute A2(2) loop models on a strip
Bibliographic record
Abstract
Abstract We study the dilute <?CDATA $A_{2}^{(2)}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msubsup><mml:mi>A</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> loop models on the geometry of a strip of width N . Two families of boundary conditions are known to satisfy the boundary Yang–Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as its T - and Y -systems, for these four boundary conditions and with a generic crossing parameter λ . For <?CDATA $\lambda/\pi$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>λ</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>π</mml:mi></mml:math> rational and thus <?CDATA $q = -\mathsf{e}^{4\mathrm{i}\lambda}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="sans-serif">e</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mrow><mml:mi mathvariant="normal">i</mml:mi></mml:mrow><mml:mi>λ</mml:mi></mml:mrow></mml:msup></mml:math> a root of unity, we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large- N expansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for small N . The present work complements a previous study (Morin-Duchesne and Pearce 2019 J. Stat. Mech. 1909) that investigated the dilute <?CDATA $A_{2}^{(2)}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msubsup><mml:mi>A</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math> loop models with periodic boundary conditions.
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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 | 0.002 | 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 | 0.000 | 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".