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

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.

💡 Research Summary

The paper presents a complete Deift‑Zhou steep‑descent analysis for a 3 × 3 matrix Riemann–Hilbert problem (RHP‑Y) that arises in quadratic Hermite–Padé approximation and multiple orthogonal polynomials. The authors focus on a “regular hard‑soft two‑edge regime”: the contour Σ consists of two active arcs Γ* and Δ* forming a compact conductor Δ = (0, x₀) with a hard edge at the origin (0) and a soft edge at a positive point x₀. Under a set of seven structural assumptions (R1)–(R7) they guarantee that the phase functions φ_Γ and φ_Δ have a non‑degenerate sign chart, that admissible lens neighborhoods exist, and that the endpoint behavior is of Bessel (hard) and Airy (soft) type.

The analysis proceeds in a modular fashion:

1. g‑function normalization – Using the equilibrium (or vector equilibrium) problem, a scalar g‑function and a constant ℓ are introduced, leading to the transformed matrix X(z)=e^{-n(g(z)−ℓ)σ₃}Y(z). The jump matrices are then expressed in terms of the analytic phase functions φ_Γ, φ_Δ.

2. Lens opening – Thanks to (R3) the real parts of the phase functions are uniformly positive on the lens lips, which yields exponentially small jumps e^{-nc} away from the endpoint disks U₀ and U_{x₀}. The factorization respects the channel‑wise structure (only (1,3) and (1,2) entries are non‑trivial).

3. Reduced outer parametrix – After lens opening the remaining central jump is independent of n and consists solely of permutation‑type jumps. The outer model N(z) solves this constant‑jump RHP, is uniquely determined by the normalization N(∞)=I, and can be written explicitly in terms of elementary algebraic functions and permutation matrices.

4. Local parametrices – Near the hard edge 0 a Bessel parametrix P^{(0)} is built from the standard Bessel functions J_α, Y_α using the local conformal coordinate f₀(z) derived from φ_Γ. Near the soft edge x₀ an Airy parametrix P^{(x₀)} is constructed from Ai, Bi with conformal coordinate f_{x₀}(z). Both parametrices match the outer solution N on the boundaries ∂U₀, ∂U_{x₀} up to O(1/n).

5. Error analysis – The error matrix R(z)=Y(z)·(global·local)^{-1} satisfies a small‑norm RHP: its jumps are either exponentially small on the lens lips or O(1/n) on the circles ∂U₀, ∂U_{x₀}. Standard L²–L^∞ estimates give R(z)=I+O(1/n) uniformly on compact subsets of ℂ\Σ_R.

6. Reconstruction and strong asymptotics – Combining the pieces yields the main result: for large n,

   • In the outer region (compact subsets of ℂ\ (Σ_R∪U₀∪U_{x₀}))
     Y_{11}(z)=e^{nG(z)}