High precision canonical Monte Carlo determination of the growth constant of square lattice trees
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Bibliographic record
Abstract
The number of lattice bond trees in the square lattice (counted modulo translations), ${t}_{n},$ is a basic quantity in lattice statistical mechanical models of branched polymers. This number is believed to have asymptotic behavior given by ${t}_{n}\ensuremath{\sim}A{\ensuremath{\lambda}}^{n}{n}^{\ensuremath{-}\ensuremath{\theta}},$ where A is an amplitude, $\ensuremath{\lambda}$ is the growth constant, and $\ensuremath{\theta}$ the entropic exponent. In this paper, we show that $\ensuremath{\lambda}$ and $\ensuremath{\theta}$ can be determined to high accuracy by using a canonical Monte Carlo algorithm; we find that $\ensuremath{\lambda}=5.1439\ifmmode\pm\else\textpm\fi{}0.0025,$ $\ensuremath{\theta}=1.014\ifmmode\pm\else\textpm\fi{}0.022,$ where the error bars are a combined $95$% statistical confidence interval and an estimated systematic error due to uncertainties in modeling corrections to scaling. If one assumes the ``exact value'' $\ensuremath{\theta}=1$ and then determines $\ensuremath{\lambda},$ then the above estimate improves to $\ensuremath{\lambda}=5.14339\ifmmode\pm\else\textpm\fi{}0.00072.$ In addition, we also determine the longest path exponent $\ensuremath{\rho}$ and the metric exponent $\ensuremath{\nu}$ from our data: $\ensuremath{\rho}=0.74000\ifmmode\pm\else\textpm\fi{}0.00062,$ $\ensuremath{\nu}=0.6437\ifmmode\pm\else\textpm\fi{}0.0035,$ with error bars similarly a combined $95$% statistical confidence interval and an estimate of the systematic error.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| 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 it