Calculation of the constant factor in the six-vertex model
Bibliographic record
Abstract
We calculate explicitly the constant factor C in the large N asymptotics of the partition function Z_N of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and ferroelectric phases. On the critical line the weights a,b,c of the model are parameterized by a parameter \alpha >1 , as a=\frac{\alpha-1}{2} , b=\frac{\alpha +1}{2} , c=1 . The asymptotics of Z_N on the critical line was obtained earlier in the paper [8] of Bleher and Liechty: Z_N=CF^{N^2}G^{\sqrt{N}}N^{1/4}(1+O(N^{-1/2})) , where F and G are given by explicit expressions, but the constant factor C>0 was not known. To calculate the constant C , we find, by using the Riemann–Hilbert approach, an asymptotic behavior of Z_N in the double scaling limit, as N and \alpha tend simultaneously to \infty in such a way that \frac{N}{\alpha}\to t\ge 0 . Then we apply the Toda equation for the tau-function to find a structural form for C , as a function of \alpha , and we combine the structural form of C and the double scaling asymptotic behavior of Z_N to calculate C .
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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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.005 | 0.002 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.005 | 0.002 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.016 | 0.003 |
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".