Random matrices with log-range correlations, and log-Sobolev inequalities
Bibliographic record
Abstract
Let X N be a symmetric N × N random matrix whose N -scaled entries are uniformly square integrable. We prove that if the entries of X N can be partitioned into independent subsets each of size o ( log N ) , then the empirical eigenvalue distribution of X N , minus its mean, converges weakly to 0 in probability; hence if the averaged empirical eigenvalue distribution converges to a law, the empirical spectral distribution converges to this limit law weakly in probability. If the entries are bounded, the convergence is almost sure; if the entries are Gaussian, we prove almost sure convergence with larger blocks of size o ( N 2 / log N ) . This significantly extends the best previously known results on convergence of eigenvalues for matrices with correlated entries, where the partition subsets are blocks and of size O ( 1 ) . We also prove the strongest known convergence results for eigenvalues of band matrices. We prove these results by developing a new log-Sobolev inequality which generalizes the second author’s introduction of mollified log-Sobolev inequalities: we show that if Y is a bounded random vector and Z is a standard normal random vector independent from Y , then the law of Y + t 1 / 2 Z satisfies a log-Sobolev inequality for all t > 0 , and we give bounds on the optimal log-Sobolev constant.
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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 | 0.005 | 0.032 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.009 | 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".