Reinforced walks in two and three dimensions
Bibliographic record
Abstract
In probability theory, reinforced walks are random walks on a lattice (or more generally a graph) that preferentially revisit neighboring 'locations' (sites or bonds) that have been visited before. In this paper, we consider walks with one-step reinforcement, where one preferentially revisits locations irrespective of the number of visits. Previous numerical simulations (A Ordemann et al 2001 Phys. Rev. E 64 046117) suggested that the site model on the lattice shows a phase transition at finite reinforcement between a random-walk-like and a collapsed phase, in both two and three dimensions. The very different mathematical structure of bond and site models might also suggest different phenomenology (critical properties, etc). We use high statistics simulations and heuristic arguments to argue that site and bond reinforcement are in the same universality class. We find broad agreement with the phase transition results of Ordemann et al , while improving their critical parameter estimates and suggesting that the phase transition in two dimensions actually occurs at zero coupling constant. We also show that a quasistatic approximation predicts the large time scaling of the end-to-end distance in the collapsed phase of both site and bond reinforcement models, in excellent agreement with simulation results.
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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 | 0.000 | 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 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".