A special set of exceptional times for dynamical random walk on $Z^2$
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Bibliographic record
Abstract
In [2] Benjamini, Haggstrom, Peres and Steif introduced the model of dynamical random walk on the $d$-dimensional lattice $Z^d$. This is a continuum of random walks indexed by a time parameter $t$. They proved that for dimensions $d=3,4$ there almost surely exist times $t$ such that the random walk at time $t$ visits the origin infinitely often, but for dimension 5 and up there almost surely do not exist such $t$. Hoffman showed that for dimension 2 there almost surely exists $t$ such that the random walk at time $t$ visits the origin only finitely many times [5]. We refine the results of [5] for dynamical random walk on $Z^2$, showing that with probability one the are times when the origin is visited only a finite number of times while other points are visited infinitely often.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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 it