Pascal's Triangle and the Kesten-McKay Law
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Bibliographic record
Abstract
The Kesten-McKay law describes the number of closed walks on a regular tree or, equivalently, the expected eigenvalues of the adjacency matrix of a random regular graph. It is a widely studied and heavily used distribution. We show that the moments of the Kesten-McKay law can be generated by the truncation of Pascal's triangle. As introduced by Kesten and later by McKay, these laws were indexed by the degree of a regular graph. However the parameter can be any positive real number even though there is no longer an associated random walk or graph. Nevertheless, we show that our triangle rule remains valid in the continuous case. Thus, we obtain a new stochastic process which we call the Kesten-McKay process. Using free independence, we give an explicit realization of the process using random matrices. Along the way we will introduce many of the standard distributions (the 'Lego blocks') of free probability. By enabling the reader to play with these distributions, we hope to entice the reader into the new world of free probability.
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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 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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