Marchenko-Pastur Theorem and Bercovici-Pata bijections for heavy-tailed\n or localized vectors
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Bibliographic record
Abstract
The celebrated Marchenko-Pastur theorem gives the asymptotic spectral\ndistribution of sums of random, independent, rank-one projections. Its main\nhypothesis is that these projections are more or less uniformly distributed on\nthe first grassmannian, which implies for example that the corresponding\nvectors are delocalized, i.e. are essentially supported by the whole canonical\nbasis. In this paper, we propose a way to drop this delocalization assumption\nand we generalize this theorem to a quite general framework, including random\nprojections whose corresponding vectors are localized, i.e. with some\ncomponents much larger than the other ones. The first of our two main examples\nis given by heavy tailed random vectors (as in a model introduced by Ben Arous\nand Guionnet or as in a model introduced by Zakharevich where the moments grow\nvery fast as the dimension grows). Our second main example is given by vectors\nwhich are distributed as the Brownian motion on the unit sphere, with localized\ninitial law. Our framework is in fact general enough to get new correspondences\nbetween classical infinitely divisible laws and some limit spectral\ndistributions of random matrices, generalizing the so-called Bercovici-Pata\nbijection.\n
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
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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