Expansion in ${\boldsymbol{n^{-1}}}$ for Percolation Critical Values on the $n$-cube and ${\boldsymbol{{\mathbb Z}^n}}$: the First Three Terms
Bibliographic record
Abstract
Let $p_c({\mathbb Q}_n)$ and $p_c({\mathbb Z}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$ -cube ${\mathbb Q}_n = \{0,1\}^n$ and on ${\mathbb Z}^n$ , respectively. Let $\Omega = n$ for ${\mathbb G} = {\mathbb Q}_n$ and $\Omega = 2n$ for ${\mathbb G} = {\mathbb Z}^n$ denote the degree of ${\mathbb G}$ . We use the lace expansion to prove that for both ${\mathbb G} = {\mathbb Q}_n$ and ${\mathbb G} = {\mathbb Z}^n$ , \[p_c({\mathbb G}) = \Omega^{-1} + \Omega^{-2} + \frac{7}{2} \Omega^{-3} + O(\Omega^{-4}).\] This extends by two terms the result $p_c({\mathbb Q}_n) = \Omega^{-1} + O(\Omega^{-2})$ of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and Slade for ${\mathbb Z}^n$ .
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.002 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.027 | 0.005 |
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".