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

TL;DR

This paper develops a detailed steepest descent analysis for a 3x3 Riemann-Hilbert problem related to Hermite-Pade approximation, providing uniform asymptotics with explicit error bounds in a two-edge regime.

Contribution

It introduces a reusable analytic scheme for asymptotic analysis of a specific Riemann-Hilbert problem under general assumptions, including explicit local parametrices.

Findings

Uniform strong asymptotics with explicit error bounds

Effective description of the solution via reduced outer and local parametrices

Application of the scheme to a two-edge regime with hard and soft edges

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.