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Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behaviour and the first Painleve' equation
Exactly Solvable and Integrable Systems · Submitted 1 Aug 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our results are: A) We found global (in t) asymptotic of recurrence coefficients and of "square-norms" for the orthogonal polynomials for different configurations of the contours of integration.
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We study the asymptotics of recurrence coefficients for monic orthogonal polynomials p_n(z) with the quartic exponential weight exp [-N (1/2 z^2 + t/4 z^4)], where t is complex. Our goals are: A) to describe the regions of different asymptotic behaviour (different genera) globally in t; B) to identify all the critical points, and; C) to study in details the asymptotics in a full neighborhood near of critical points (double scaling limit), including at and near the poles of Painleve' I solutions y(v) that are known to provide the leading correction term in this limit. Our results are: A) We found global (in t) asymptotic of recurrence coefficients and of "square-norms" for the orthogonal polynomials for different configurations of the contours of integration. Special code was developed to analyze all possible cases. B) In addition to the known critical point t_0=- 1/ 12, we found new critical points t_1=1/15 and t_2=1/4. C) We derived the leading order behavior of the recurrence coefficients (together with the error estimates) at and around the poles of y(v) near the critical points t_0,t_1 in what we called the triple scaling limit. We proved that the recurrence coefficients have unbounded "spikes" near the poles of y(v) and calculated the "universal" shape of these spikes for different cases. The nonlinear steepest descent method for Riemann-Hilbert Problem (RHP) is the main technique used in the paper. We note that the RHP near the critical points is very similar to the RHP describing the semiclassical limit of the focusing NLS near the point of gradient catastrophe that the authors solved previously.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the asymptotics of recurrence coefficients for monic orthogonal polynomials p_n(z) with the quartic exponential weight exp [-N (1/2 z^2 + t/4 z^4)], where t is complex. | contribution 0.52 | 0.85 | 0.04 | asserted 0.92 |
| s1 | Our goals are: A) to describe the regions of different asymptotic behaviour (different genera) globally in t; B) to identify all the critical points, and; C) to study in details the asymptotics in a full neighborhood near of critical points (double scaling limit), including at and near the poles of Painleve' I solutions y(v) that are known to provide the leading correction term in this limit. | contribution 0.68 | 0.23 | 0.05 | asserted 0.67 |
| s2 | Our results are: A) We found global (in t) asymptotic of recurrence coefficients and of "square-norms" for the orthogonal polynomials for different configurations of the contours of integration. | result 0.99 | 0.96 | 0.38 | asserted 1.00 |
| s3 | Special code was developed to analyze all possible cases. | method 0.98 | 0.89 | 0.02 | asserted 1.00 |
| s4 | B) In addition to the known critical point t_0=- 1/ 12, we found new critical points t_1=1/15 and t_2=1/4. | result 0.99 | 0.95 | 0.12 | asserted 1.00 |
| s5 | C) We derived the leading order behavior of the recurrence coefficients (together with the error estimates) at and around the poles of y(v) near the critical points t_0,t_1 in what we called the triple scaling limit. | result 0.97 | 0.97 | 0.18 | asserted 1.00 |
| s6 | We proved that the recurrence coefficients have unbounded "spikes" near the poles of y(v) and calculated the "universal" shape of these spikes for different cases. | result 1.00 | 0.96 | 0.21 | asserted 1.00 |
| s7 | The nonlinear steepest descent method for Riemann-Hilbert Problem (RHP) is the main technique used in the paper. | method 1.00 | 0.38 | 0.03 | asserted 1.00 |
| s8 | We note that the RHP near the critical points is very similar to the RHP describing the semiclassical limit of the focusing NLS near the point of gradient catastrophe that the authors solved previously. | interpretation 0.51 | 0.44 | 0.03 | asserted 0.97 |
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The paper
Original abstract
We study the asymptotics of recurrence coefficients for monic orthogonal polynomials p_n(z) with the quartic exponential weight exp [-N (1/2 z^2 + t/4 z^4)], where t is complex. Our goals are: A) to describe the regions of different asymptotic behaviour (different genera) globally in t; B) to identify all the critical points, and; C) to study in details the asymptotics in a full neighborhood near of critical points (double scaling limit), including at and near the poles of Painleve' I solutions y(v) that are known to provide the leading correction term in this limit. Our results are: A) We found global (in t) asymptotic of recurrence coefficients and of "square-norms" for the orthogonal polynomials for different configurations of the contours of integration. Special code was developed to analyze all possible cases. B) In addition to the known critical point t_0=- 1/ 12, we found new critical points t_1=1/15 and t_2=1/4. C) We derived the leading order behavior of the recurrence coefficients (together with the error estimates) at and around the poles of y(v) near the critical points t_0,t_1 in what we called the triple scaling limit. We proved that the recurrence coefficients have unbounded "spikes" near the poles of y(v) and calculated the "universal" shape of these spikes for different cases. The nonlinear steepest descent method for Riemann-Hilbert Problem (RHP) is the main technique used in the paper. We note that the RHP near the critical points is very similar to the RHP describing the semiclassical limit of the focusing NLS near the point of gradient catastrophe that the authors solved previously.
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How the literature uses it
10 arXiv papers cite this in 29 sentences
Papers citing it · 20 sentences without a confident label
Builds on it 1 paper
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In this paper we continue (see [ref] ) our study of monic polynomials $π_n(z)$, orthogonal with respect to the quartic exponential weight $e^-N f(z,t)$, where $f(z,t)=z^2 + t z^4$, $t∈$ and $N∈$, and the contour of integration consists of four rays in the directions of the exponential decay of $e^-N f(z,t)$.
On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight · Introduction and main results · 0.86
Uses its method or tool 1 paper
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Our goal is to rigorously justify the asymptotic phase portrait for the Generic case (case (a1) in [ref] ) using the principle of continuation in the parameter space, developed in [ref] and [ref] .
On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight · Introduction and main results · 0.98
Compares with it 1 paper
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It is also worth noticing that in terms of the variable $w=t^-3/2$ the curve ([ref]) looks quite similar to the genus 0 breaking curve found in [ref] for the family of orthogonal polynomials associated to the quartic potential [equation] However, the curve in [ref] is only symmetric with respect the real $w$-axis, while the curve for the cubic model is symmetric with respect both real and imaginary axes.
Determination of S-curves with applications to the theory of nonhermitian orthogonal polynomials · Asymptotic zero distributions of orthogonal polynomials · 1.00
Cites it as background 4 papers
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For the benefit of the reader, we repeat here some facts from [ref] that are required for the exposition.
On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight · Introduction and main results · 0.95
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In [ref] , orthogonal polynomials with respect to complex weights of the form $e^-nV(x)$ were considered, with $V$ quartic symmetric with complex-valued leading coefficient.
Critical asymptotic behavior for the Korteweg-de Vries equation and in random matrix theory · Example: quartic external field · 0.89
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The case of a quartic polynomial potential is studied in detail in [ref] .
Zero distribution of complex orthogonal polynomials with respect to exponential weights · Introduction · 0.94
What it cites · 8 sentences without a confident label
Uses the method or tool of 2 papers
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Computation of the recurrence coefficients: Following [ref] , we find the expansion of the matrix $Φ(z)=E(z) Ψ_1(z)$ at $z=∞$:
Universality for the focusing nonlinear Schroedinger equation at the gradient catastrophe point: Rational breathers and poles of the tritronquee solution to Painleve I · The symmetric case: proof of Theorem [ref] · 0.91
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We rather explain briefly the algorithm used to investigate the phase portraits; in [ref] an algorithm to find numerically "Boutroux curves" was explained.
Boutroux curves with external field: equilibrium measures without a minimization problem · Phase portraits or distribution of genera in the complex $t$-plane · 0.96
Compares with 1 paper
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The parametrix $𝒫_1$ coincides with the parametrix considered in [ref] up to conjugation by $_2$.
Universality for the focusing nonlinear Schroedinger equation at the gradient catastrophe point: Rational breathers and poles of the tritronquee solution to Painleve I · Asymptotics away from the poles · 0.99
Cites as background 1 paper
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A general discussion of contour deformations once the appropriate $g$–function has been found can be read in [ref] and [ref] ; we give here a brief sketch.
Boutroux curves with external field: equilibrium measures without a minimization problem · Contour deformation. · 0.92
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