Finite Rank Perturbations of Random Matrices and their Continuum Limits
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Bibliographic record
Abstract
We study Gaussian sample covariance matrices with population covariance a bounded-rank perturbation of the identity, as well as Wigner matrices with bounded-rank additive perturbations. The top eigenvalues are known to exhibit a phase transition in the large size limit: with weak perturbations they follow Tracy-Widom statistics as in the unperturbed case, while above a threshold there are outliers with independent Gaussian fluctuations. Baik, Ben Arous and Péché (2005) described the transition in the complex case and conjectured a similar picture in the real case, the latter of most relevance to high-dimensional data analysis. Resolving the conjecture, we prove that in all cases the top eigenvalues have a limit near the phase transition. Our starting point is the work of Rámirez, Rider and Virág (2006) on the general beta random matrix soft edge. For rank one perturbations, a modified tridiagonal form converges to the known random Schrödinger operator on the half-line but with a boundary condition that depends on the perturbation. For general finite-rank perturbations we develop a new band form; it converges to a limiting operator with matrix-valued potential. The low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. Their laws are also characterized in terms of a diffusion related to Dyson's Brownian motion and in terms of a linear parabolic PDE. We offer a related heuristic for the supercritical behaviour and rigorously treat the supercritical asymptotics of the ground state of the limiting operator. In a further development, we use the PDE to make the first explicit connection between a general beta characterization and the celebrated Painlevé representations of Tracy and Widom (1993, 1996). In particular, for beta = 2,4 we give novel proofs of the latter. Finally, we report briefly on evidence suggesting that the PDE provides a stable, even efficient method for numerical evaluation of the Tracy-Widom distributions, their general beta analogues and the deformations discussed and introduced here. This thesis is based in part on work to be published jointly with Bálint Virág.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| 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.
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