Random mappings with Ewens cycle structure
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Bibliographic record
Abstract
In this paper we consider a random mapping, Tn,θ, of the finite set {1, 2, ..., n} into itself for which the digraph representation Ĝn,θ is constructed by: (1) selecting a random number, Ln, of cyclic vertices, (2) constructing a uniform random forest of size n with the selected cyclic vertices as roots, and (3) forming ‘cycles’ of trees by applying to the selected cyclic vertices a random permutation with cycle structure given by the Ewens sampling formula with parameter θ. We investigate kn,θ, the size of a ‘typical’ component of Ĝn,θ, and we obtain the asymptotic distribution of kn,θ conditioned on Ln = m(n). As an application of our results, we show in Section 3 that provided Ln is of order much larger than √ n, then the joint distribution of the normalized order statistics of the component sizes of Ĝn,θ converges to the Poisson-Dirichlet(θ) distribution as n→∞. ∗Actuarial Mathematics and Statistics Department and The Maxwell Institute for Mathematical Sciences, Heriot–Watt University, Edinburgh EH14 4AS, UK; email: J.Hansen@hw.ac.uk †Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznan, Poland; email: jaworski@amu.edu.pl ‡J. Jaworski acknowledges the support by the Marie Curie Intra-European Fellowship No. 236845 (RANDOMAPP) within the 7th European Community Framework Programme and by National Science Centre DEC-2011/01/B/ST1/03943.
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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 | 0.000 |
| 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 | 0.001 |
| Open science | 0.001 | 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