High-dimensional near-critical percolation and the torus plateau
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Bibliographic record
Abstract
We consider percolation on Zd and on the d-dimensional discrete torus, in dimensions d≥11 for the nearest-neighbour model and in dimensions d>6 for spread-out models. For Zd we employ a wide range of techniques and previous results to prove that there exist positive constants c and C such that the slightly subcritical two-point function and one-arm probabilities satisfy Ppc−ε(0↔x)≤C‖x‖d−2e−cε1/2‖x‖,cr2e−Cε1/2r≤Ppc−ε(0↔∂[−r,r]d)≤Cr2e−cε1/2r. Using this, we prove that throughout the critical window the torus two-point function has a “plateau,” meaning that it decays for small x as ‖x‖−(d−2) but for large x is essentially constant and of order V−2/3 where V is the volume of the torus. The plateau for the two-point function leads immediately to a proof of the torus triangle condition, which is known to have many implications for the critical behaviour on the torus, and also leads to a proof that the critical values on the torus and on Zd are separated by a multiple of V−1/3. The torus triangle condition and the size of the separation of critical points have been proved previously, but our proofs are different and are direct consequences of the bound on the Zd two-point function. In particular, we use results derived from the lace expansion on Zd, but in contrast to previous work on high-dimensional torus percolation, we do not need or use a separate torus lace expansion.
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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 | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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