Random walks with fractally correlated traps: Stretched exponential and power-law survival kinetics
Bibliographic record
Abstract
We consider the survival probability $f(t)$ of a random walk with a constant hopping rate $w$ on a host lattice of fractal dimension $d$ and spectral dimension ${d}_{s}\ensuremath{\le}2$, with spatially correlated traps. The traps form a sublattice with fractal dimension ${d}_{a}<d$ and are characterized by the absorption rate ${w}_{a}$ which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps (${w}_{a}\ensuremath{\ll}w$), we find that $f(t)$ can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent $\ensuremath{\alpha}=1\ensuremath{-}(d\ensuremath{-}{d}_{a})/{d}_{w}$, where ${d}_{w}$ is the random walk dimension of the host lattice. At the end of this initial stage there occurs a crossover to power-law kinetics $f(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\alpha}}$ with the same exponent $\ensuremath{\alpha}$ as for the stretched exponential regime. For strong absorption ${w}_{a}\ensuremath{\gtrsim}w$, including the limit of perfect traps ${w}_{a}\ensuremath{\rightarrow}\ensuremath{\infty}$, the stretched exponential regime is absent and the decay of $f(t)$ follows, after a short transient, the aforementioned power law for all times.
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 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".