Critical exponents for the homology of Fortuin-Kasteleyn clusters on a torus
Bibliographic record
Abstract
A Fortuin-Kasteleyn cluster on a torus is said to be of type {a,b},a,b in Z , if it is possible to draw a curve belonging to the cluster that winds a times around the first cycle of the torus as it winds -b times around the second. Even though the Q -Potts models make sense only for Q integers, they can be included into a family of models parametrized by beta = square root of Q for which the Fortuin-Kasteleyn clusters can be defined for any real beta(0,2] . For this family, we study the probability pi({a,b}) of a given type of clusters as a function of the torus modular parameter tau=tau(r)+itau(i). We compute the asymptotic behavior of some of these probabilities as the torus becomes infinitely thin. For example, the behavior of pi({1,0}) is studied for tau(i) --> infinity . Exponents describing these behaviors are defined and related to weights h(r,s) of the extended Kac table for r and s integers, but also half-integers. Numerical simulations are also presented. Possible relationship with recent works and conformal loop ensembles is discussed.
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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 | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.004 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".