A Riemann–Hilbert approach to asymptotic analysis of Toeplitz+Hankel determinants II

Abstract In this article, we further develop the Riemann–Hilbert formalism introduced in our earlier work [Gharakhloo and Its, SIGMA 16 (2020), 100] for the asymptotic analysis of Toeplitz+Hankel determinants with distinct Toeplitz and Hankel symbols. In that work, we showed that the associated Riemann–Hilbert problem can be analyzed by the Deift–Zhou nonlinear steepest descent method under a special restriction on the winding numbers of the symbols. This restriction excludes several natural cases, including the zero–winding configuration for both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ϕ</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> </mml:math> . The main goal of the present paper is to extend the asymptotic analysis of Toeplitz+Hankel determinants to a broader class of winding–number configurations. As an application, we study the case associated with the Ising model on the zig–zag layered half–plane with critical boundary magnetic field, for which the winding numbers of the Toeplitz and Hankel symbols are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , respectively. In this setting, we compute the asymptotics of the norms of the corresponding system of orthogonal polynomials.