Spacing distributions for point processes on a regular fractal
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Bibliographic record
Abstract
The homogeneous Poisson point process in Rd (denoted by Pd) is a basic model of stochastic geometry and modern statistical physics. Using ideas from fractal geometry, geometrical statistics, and random matrix theory, we introduce the model of random points on a self-similar fractal as a model of intermediate statistics, in the sense that the interpoint spacing statistics of the model are intermediate between those of P1 and P2 when the fractal dimension is in between 1 and 2, and intermediate between those of P2 and P3 when the fractal dimension is in between 2 and 3, and so on. We also introduce the idea of using a continuous family of such models to interpolate between P1 and P2 and thereby effectuate crossover transitions between P1 statistics and P2 statistics. We first derive the kth-nearest-neighbor spacing distribution for the general model, and then study the interpoint spacing statistics of several realizations of the model involving Sierpinski fractals in R2 and R3. We also study a realization of a continuous interpolation between P1 and P2, in particular a continuous interpolation between a point process on a line and a point process on a plane-filling curve, using the continuous family of self-similar Koch curves in R2. In the latter study, we specifically analyze the second-nearest-neighbor interpoint spacing statistics, which undergo a crossover transition between semi-Poisson and Ginibre statistics.
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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 | 0.000 | 0.002 |
| 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 it