Boundary conditions and universal finite‐size scaling for the hierarchical |φ|4$|\varphi |^4$ model in dimensions 4 and higher
Bibliographic record
Abstract
Abstract We analyse and clarify the finite‐size scaling of the weakly‐coupled hierarchical ‐component model for all integers in all dimensions , for both free and periodic boundary conditions. For , we prove that for a volume of size with periodic boundary conditions the infinite‐volume critical point is an effective finite‐volume critical point, whereas for free boundary conditions the effective critical point is shifted smaller by an amount of order . For both boundary conditions, the average field has the same non‐Gaussian limit within a critical window of width around the effective critical point, and in that window we compute the universal scaling profile for the susceptibility. In contrast, and again for both boundary conditions, the average field has a massive Gaussian limit when above the effective critical point by an amount . In particular, at the infinite‐volume critical point the susceptibility scales as for periodic boundary conditions and as for free boundary conditions. We identify a mass generation mechanism for free boundary conditions that is responsible for this distinction and which we believe has wider validity, in particular to Euclidean (non‐hierarchical) models on in dimensions . For we prove a similar picture with logarithmic corrections. Our analysis is based on the rigorous renormalisation group method of Bauerschmidt, Brydges and Slade, which we improve and extend.
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How this classification was reachedexpand
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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".