Beta ensembles, stochastic Airy spectrum, and a diffusion
Bibliographic record
Abstract
We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schrödinger operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus StartFraction d squared Over d x squared EndFraction plus x plus StartFraction 2 Over StartRoot beta EndRoot EndFraction b prime Subscript x"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mfrac> <mml:msup> <mml:mi>d</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mi> β </mml:mi> </mml:msqrt> </mml:mfrac> <mml:msubsup> <mml:mi>b</mml:mi> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-variant" mathvariant="normal"> ′ </mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">-\frac {d^2}{dx^2} + x + \frac {2}{\sqrt {\beta }} b_x^{\prime }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> restricted to the positive half-line, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b prime Subscript x"> <mml:semantics> <mml:msubsup> <mml:mi>b</mml:mi> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-variant" mathvariant="normal"> ′ </mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">b_x^{\prime }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is white noise. In doing so we extend the definition of the Tracy-Widom( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi> β </mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) distributions to all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converges to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem.
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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 | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 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".