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Uniformity of Strong Asymptotics in Angelesco Systems

Maxim L. Yattselev

TL;DR

This work develops a uniform, strong asymptotic theory for type II and type I multiple orthogonal polynomials associated with Angelesco systems. The authors construct a detailed Riemann–Hilbert framework, including a three-sheeted surface and conformal maps, to obtain asymptotics that are uniform in the minimum of the two multi-indices. They build global and local parametrices, analyze several edge regimes (hard, sliding soft, and critical soft edges), and prove uniform error bounds, while also deriving differential relations for edge parameters and recurrence coefficients. The results extend previous diagonal- and subsequence-based analyses by removing the need for sub-sequence limits and establishing uniformity across all admissible index sequences, with potential impact on stability of recurrence relations and zero-distribution descriptions. The methods yield explicit expressions for the leading factors, recurrence coefficients, and edge-dynamics, contributing to a deeper understanding of the interaction between geometry, potential theory, and asymptotics in Angelesco systems.

Abstract

Let $μ_1$ and $μ_2$ be two complex-valued Borel measures on the real line such that $\operatorname{supp} μ_1 =[α_1,β_1] < \operatorname{supp} μ_2 =[α_2,β_2]$ and ${\rm d}μ_i(x) = -ρ_i(x){\rm d}x/2π{\rm i}$, where $ρ_i(x)$ is the restriction to $[α_i,β_i]$ of a function non-vanishing and holomorphic in some neighborhood of $[α_i,β_i]$. Strong asymptotics of multiple orthogonal polynomials is considered as their multi-indices $(n_1,n_2)$ tend to infinity in both coordinates. The main goal of this work is to show that the error terms in the asymptotic formulae are uniform with respect to $\min\{n_1,n_2\}$.

Uniformity of Strong Asymptotics in Angelesco Systems

TL;DR

This work develops a uniform, strong asymptotic theory for type II and type I multiple orthogonal polynomials associated with Angelesco systems. The authors construct a detailed Riemann–Hilbert framework, including a three-sheeted surface and conformal maps, to obtain asymptotics that are uniform in the minimum of the two multi-indices. They build global and local parametrices, analyze several edge regimes (hard, sliding soft, and critical soft edges), and prove uniform error bounds, while also deriving differential relations for edge parameters and recurrence coefficients. The results extend previous diagonal- and subsequence-based analyses by removing the need for sub-sequence limits and establishing uniformity across all admissible index sequences, with potential impact on stability of recurrence relations and zero-distribution descriptions. The methods yield explicit expressions for the leading factors, recurrence coefficients, and edge-dynamics, contributing to a deeper understanding of the interaction between geometry, potential theory, and asymptotics in Angelesco systems.

Abstract

Let and be two complex-valued Borel measures on the real line such that and , where is the restriction to of a function non-vanishing and holomorphic in some neighborhood of . Strong asymptotics of multiple orthogonal polynomials is considered as their multi-indices tend to infinity in both coordinates. The main goal of this work is to show that the error terms in the asymptotic formulae are uniform with respect to .

Paper Structure

This paper contains 37 sections, 16 theorems, 200 equations, 4 figures.

Table of Contents

  1. Main results
  2. Multiple orthogonal polynomials
  3. Angelesco systems
  4. Riemann surfaces
  5. Conformal maps
  6. Main asymptotic terms
  7. Main results
  8. Model local parametrices
  9. Hard edge
  10. Sliding soft edge
  11. Critical soft edge
  12. Conformal maps
  13. Conformal maps
  14. Additional properties
  15. Polygons of conformality
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $\mu_1$ and $\mu_2$ be as in mus, where $\rho_1(x)$ and $\rho_2(x)$ are the restrictions to $\Delta_1$ and $\Delta_2$, respectively, of non-vanishing functions analytic in some neighborhood of the corresponding interval. Set where $c({\vec{n}}) = n_1/|{\vec{n}}|$ and $\gamma_{{\vec{n}},i}$ are the normalizing constants, $i\in\{1,2\}$. Let $P_{\vec{n}}(z)$ be the type II multiple orthogonal po

Figures (4)

  • Figure 1: Realization of the surface $\boldsymbol{\mathfrak{S}}_c$ (the notation is meant to emphasize that pushing effect happens on one interval only, i.e., $\beta_{c,1}<\beta_1$ and therefore $\alpha_{c,2}=\alpha_2$ for this value of $c$).
  • Figure 2: Polygons $\partial U_{c,\beta_1}$; panel (a) $c\in[c_r,c^*)$; panel (b) $c\in(0,c_r)$.
  • Figure 3: The boundaries $\partial U_{{\vec{n}},\alpha_1}$ and $\partial U_{{\vec{n}},\beta_{{\vec{n}},1}}$, arcs $\Gamma_{{\vec{n}},1}^\pm$, and domains $\Omega_{{\vec{n}},1}^\pm$ (shaded).
  • Figure 4: Lens $\Sigma_{{\vec{n}},\delta}$ consisting of two connected components $\Sigma_{{\vec{n}},\delta,1}$ (the left one) and $\Sigma_{{\vec{n}},\delta,2}$ (the right one).

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 14 more