End-to-end distance from the Green’s function for a hierarchical self-avoiding walk in four dimensions
Bibliographic record
Abstract
In [BEI92] we introduced a Levy process on a hierarchical lattice which is four 1 dimensional, in the sense that the Green’s function for the process equals |x | 2. If the process is modified so as to be weakly self-repelling, it was shown that at the critical killing rate (mass-squared) βc, the Green’s function behaves like the free one. Now we analyze the end-to-end distance of the model and show that its expected value grows as a constant times √ T log 1 8 T 1 + O, which is log log T log T the same law as has been conjectured for self-avoiding walks on the simple cubic lattice Z 4. The proof uses inverse Laplace transforms to obtain the end-to-end distance from the Green’s function, and requires detailed properties of the Green’s function throughout a sector of the complex β plane. These estimates are derived in a companion paper [BI02]. Research supported by NSF Grant DMS-9706166Contents
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".