Secular coefficients and the holomorphic multiplicative chaos
Bibliographic record
Abstract
We study the secular coefficients of N×N random unitary matrices UN drawn from the Circular β-Ensemble which are defined as the coefficients of {zn} in the characteristic polynomial det(1−zUN∗). When β>4, we obtain a new class of limiting distributions that arise when both n and N tend to infinity simultaneously. We solve an open problem of Diaconis and Gamburd (Electron. J. Combin. 11 (2004/06) 2) by showing that, for β=2, the middle coefficient of degree n=⌊N2⌋ tends to zero as N→∞. We show how the theory of Gaussian multiplicative chaos (GMC) plays a prominent role in these problems and in the explicit description of the obtained limiting distributions. We extend the remarkable magic square formula of (Electron. J. Combin. 11 (2004/06) 2) for the moments of secular coefficients to all β>0 and analyse the asymptotic behaviour of the moments. We obtain estimates on the order of magnitude of the secular coefficients for all β>0, and we prove these estimates are sharp when β≥2 and N is sufficiently large with respect to n. These insights motivated us to introduce a new stochastic object associated with the secular coefficients, which we call Holomorphic Multiplicative Chaos (HMC). Viewing the HMC as a random distribution, we prove a sharp result about its regularity in an appropriate Sobolev space. Our proofs expose and exploit several novel connections with other areas, including random permutations, Tauberian theorems and combinatorics.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 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".