Beta ensembles, stochastic Airy spectrum, and a diffusion
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
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.
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.
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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