A complete convergence theorem for the q-voter model and other voter model perturbations in two dimensions
Bibliographic record
Abstract
The q-voter model is a spin-flip system in which the rate of flipping to type i is given by the qth power of the proportion of nearest neighbours in type i for i=0,1. If q=1 it reduces to the classical voter model. We show that in the critical 2-dimensional case, for q<1 and close enough to 1, for any initial state as t→∞ the system converges weakly to a mixture of all 0’s, all 1’s, and a unique invariant law which contains infinitely many sites of both types. This follows as a special case of a general theorem which proves a similar “complete convergence theorem” for cancellative, monotone, finite range voter model perturbations on Z2 providing a certain parameter, Θ3, is strictly positive. Similar results follow for the affine and geometric voter models and Lotka-Volterra models, all for parameter values close to that giving the voter model. This kind of asymptotic behavior is quite different from that of the 2-dimensional voter model itself, which undergoes clustering, and converges to a mixture of all 0’s and all 1’s. The above parameter Θ3 has an explicit expression in terms of asymptotic coalescing probabilities of 2-dimensional random walk, and we give a rather simple sufficient condition for it to be strictly positive. An important step in the proof is to establish weak convergence of the rescaled spin-flip systems to super-Brownian motion with drift Θ3. In fact, a convergence result is proved under weaker hypotheses which includes all known such results for 2-dimensional voter model perturbations and a number of new ones, including a rescaled limit theorem for the q-voter model where q↑1 with the rescaling.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.006 | 0.020 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.004 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.011 | 0.001 |
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".