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

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.