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Strong Asymptotics for a 3x3 Riemann-Hilbert Problem in a Regular Hard-Soft Two-Edge Regime

Artur Kandaian

TL;DR

The paper develops a complete Deift–Zhou steepest descent analysis for a $3\times3$ Riemann–Hilbert problem arising from Hermite–Padé approximation and multiple orthogonality in a regular hard/soft two-edge regime. By separating input data (equilibrium sign chart and endpoint types) from the universal analytic steps, it constructs a permutation-type outer parametrix and local Bessel and Airy parametrices, then shows the resulting error problem is small-norm with $R=I+O(1/n)$. This yields uniform strong asymptotics for the $(1,1)$-entry $Y_{11}$ in all regions, namely $Y_{11}(z)=e^{nG(z)}(N_{11}(z)+O(1/n))$ away from endpoints and analogous hard/soft edge descriptions near $0$ and $x_0$, with explicit matching on the boundaries of endpoint disks. The approach provides a modular, reusable framework for $3\times3$ hard/soft two-edge problems in Hermite–Padé and MOP settings, enabling edge universality statements and facilitating application to concrete models with permutation-type outer data.

Abstract

We develop a complete Deift-Zhou steepest descent analysis for a 3x3 matrix Riemann-Hilbert problem arising in quadratic Hermite-Pade approximation and multiple orthogonality. We focus on a regular two-edge regime with a hard edge at 0 and a soft edge at x0. Under natural geometric and analytic assumptions ensuring a nondegenerate sign structure of the associated phase functions, the standard lens-opening mechanism applies. The analysis is organized as a reusable scheme: once the equilibrium/sign-chart input is verified (assumptions R1-R7), the remaining steps are purely analytic. As a result, the solution is described in terms of a reduced outer parametrix with permutation-type jumps, complemented by Bessel- and Airy-type local parametrices at the endpoints. We obtain uniform strong asymptotics for the top-left entry, with an explicit error bound of order 1/n outside the endpoint neighborhoods.

Strong Asymptotics for a 3x3 Riemann-Hilbert Problem in a Regular Hard-Soft Two-Edge Regime

TL;DR

The paper develops a complete Deift–Zhou steepest descent analysis for a Riemann–Hilbert problem arising from Hermite–Padé approximation and multiple orthogonality in a regular hard/soft two-edge regime. By separating input data (equilibrium sign chart and endpoint types) from the universal analytic steps, it constructs a permutation-type outer parametrix and local Bessel and Airy parametrices, then shows the resulting error problem is small-norm with . This yields uniform strong asymptotics for the -entry in all regions, namely away from endpoints and analogous hard/soft edge descriptions near and , with explicit matching on the boundaries of endpoint disks. The approach provides a modular, reusable framework for hard/soft two-edge problems in Hermite–Padé and MOP settings, enabling edge universality statements and facilitating application to concrete models with permutation-type outer data.

Abstract

We develop a complete Deift-Zhou steepest descent analysis for a 3x3 matrix Riemann-Hilbert problem arising in quadratic Hermite-Pade approximation and multiple orthogonality. We focus on a regular two-edge regime with a hard edge at 0 and a soft edge at x0. Under natural geometric and analytic assumptions ensuring a nondegenerate sign structure of the associated phase functions, the standard lens-opening mechanism applies. The analysis is organized as a reusable scheme: once the equilibrium/sign-chart input is verified (assumptions R1-R7), the remaining steps are purely analytic. As a result, the solution is described in terms of a reduced outer parametrix with permutation-type jumps, complemented by Bessel- and Airy-type local parametrices at the endpoints. We obtain uniform strong asymptotics for the top-left entry, with an explicit error bound of order 1/n outside the endpoint neighborhoods.
Paper Structure (70 sections, 10 theorems, 79 equations)

This paper contains 70 sections, 10 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main result (informal overview)
  4. Outline of the paper
  5. Related work and context
  6. Problem Setup and Main Results
  7. A $3\times 3$ Riemann—Hilbert problem
  8. Multiple Laguerre polynomials (first kind).
  9. Riemann—Hilbert problem for $Y$.
  10. Hard/soft two-edge picture.
  11. Relation to the abstract assumptions (R1)—(R7).
  12. How to verify (R1)—(R7) in practice (minimal).
  13. Where (R1)—(R7) come from in practice.
  14. Regular regime assumptions
  15. Main theorem and consequences
  16. ...and 55 more sections

Key Result

Theorem 2.1

Assume (R1)—(R7). Then, as $n\to\infty$, the entry $Y_{11}$ admits the following asymptotic descriptions. Moreover, the associated error matrix $R$ satisfies uniformly for $z$ in compact subsets of $\mathbb{C}\setminus\Sigma_R$.

Theorems & Definitions (26)

  • Theorem 2.1: Strong asymptotics in the regular regime
  • Corollary 2.2: Local edge behavior: Bessel and Airy scaling
  • Lemma 3.1: Sign chart for the phases
  • Remark 3.2
  • Remark 4.1
  • Lemma 4.2: Lens decay
  • Remark 5.2
  • Proposition 5.3: Solvability and uniqueness
  • proof
  • Remark 5.4: One-sentence pointer to Appendix A
  • ...and 16 more