On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight
Bibliographic record
Abstract
We study the asymptotics of recurrence coefficients for monic orthogonal polynomials π n (z) with the quartic exponential weight exp[-N ( 1 2 z 2 + 1 4 tz 4 )], where t ∈ C and N ∈ N, N → ∞.Our goal is to describe these asymptotic behaviors globally for t ∈ C in different regions.We also describe the "breaking" curves separating these regions, and discuss their special (critical) points.All these pieces of information combined provide the global asymptotic "phase portrait" of the recurrence coefficients of π n (z), which was studied numerically in [Constr.Approx.41 (2015), 529-587, arXiv:1108.0321].The main goal of the present paper is to provide a rigorous framework for the global asymptotic portrait through the nonlinear steepest descent analysis (with the g-function mechanism) of the corresponding Riemann-Hilbert problem (RHP) and the continuation in the parameter space principle.The latter allows to extend the nonlinear steepest descent analysis from some parts of the complex t-plane to all noncritical values of t.We also provide explicit solutions for recurrence coefficients in terms of the Riemann theta functions.The leading order behaviour of the recurrence coefficients in the full scaling neighbourhoods the critical points (double and triple scaling limits) was obtained in [Constr.Approx.41 (2015), 529-587, arXiv:1108.0321] and [Asymptotics of complex orthogonal polynomials on the cross with varying quartic weight: critical point behaviour and the second Painlevé transcendents, in preparation].
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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 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) | 0.003 | 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".