On the Tracy-Widom <sub>β</sub> Distribution for β=6
Bibliographic record
Abstract
We study the Tracy-Widom distribution function for Dyson's β-ensemble with β = 6.The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Virág equation.The latter is a linear PDE which describes the Tracy-Widom functions corresponding to general values of β.Using his Lax pair, Rumanov derives an explicit formula for the Tracy-Widom β = 6 function in terms of the second Painlevé transcendent and the solution of an auxiliary ODE.Rumanov also shows that this formula allows him to derive formally the asymptotic expansion of the Tracy-Widom function.Our goal is to make Rumanov's approach and hence the asymptotic analysis it provides rigorous.In this paper, the first one in a sequel, we show that Rumanov's Lax-pair can be interpreted as a certain gauge transformation of the standard Lax pair for the second Painlevé equation.This gauge transformation though contains functional parameters which are defined via some auxiliary nonlinear ODE which is equivalent to the auxiliary ODE of Rumanov's formula.The gauge-interpretation of Rumanov's Lax-pair allows us to highlight the steps of the original Rumanov's method which needs rigorous justifications in order to make the method complete.We provide a rigorous justification of one of these steps.Namely, we prove that the Painlevé function involved in Rumanov's formula is indeed, as it has been suggested by Rumanov, the Hastings-McLeod solution of the second Painlevé equation.The key issue which we also discuss and which is still open is the question of integrability of the auxiliary ODE in Rumanov's formula.We note that this question is crucial for the rigorous asymptotic analysis of the Tracy-Widom function.We also notice that our work is a partial answer to one of the problems related to the β-ensembles formulated by Percy Deift during the June 2015 Montreal Conference on integrable systems.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".