Asymptotic freeness of random permutation matrices with restricted cycle lengths
Bibliographic record
Abstract
Let A1, A2, . . ., As be a finite sequence of (not necessarily disjoint, or even distinct) non-empty sets of positive integers such that each Ar either is a finite set or satisfies j∈N\Ar 1 j < ∞.It is shown that an independent family U1, U2, . . ., Us of uniformly distributed random N × N permutation matrices with cycle lengths restricted to A1, A2, . . ., As, respectively, converges in * -distribution as N → ∞ to a * -free family u1, u2, . . ., us of noncommutative random variables, where each ur is a (max Ar)-Haar unitary (if Ar is a finite set) or a Haar unitary (if Ar is an infinite set).Under the additional assumption that each of the sets A1, A2, . . ., As either consists of a single positive integer or is infinite, it is shown that the convergence in * -distribution actually holds almost surely.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.033 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".