High precision canonical Monte Carlo determination of the growth constant of square lattice trees
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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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 | 0.003 | 0.013 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".