A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants II

2025/09/15 by Roozbeh Gharakhloo, Alexander Its, Gharakhloo, Roozbeh +1
Mathematics · #15B05 #30E15 #30E25 #41A60 #42C05 #82B20 #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2509.12345

openalex publication_date 2025/09/15 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

In this article, we continue the development of the Riemann-Hilbert formalism for studying the asymptotics of Toeplitz+Hankel determinants with non-identical symbols, which we initiated in \citeGI. In \citeGI, we showed that the Riemann-Hilbert problem we formulated admits the Deift-Zhou nonlinear steepest descent analysis, but with a special restriction on the winding numbers of the associated symbols. In particular, the most natural case, namely zero winding numbers, is not allowed. A principal goal of this paper is to develop a framework that extends the asymptotic analysis of Toeplitz+Hankel determinants to a broader range of winding-number configurations. As an application, we consider the case in which the winding numbers of the Szegő-type Toeplitz and Hankel symbols are zero and one, respectively, and compute the asymptotics of the norms of the corresponding system of orthogonal polynomials.

Citations

• Bordered and Framed Toeplitz and Hankel Determinants with Applications to Integrable Probability
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• Algebraic aspects of increasing subsequences
• On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations
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Related

• On the asymptotics of a Toeplitz determinant with singularities
• Szegö via Jacobi
• Toeplitz and Hankel determinants with singularities: announcement of results
• Asymptotic formulas for determinants of a special class of Toeplitz + Hankel matrices
• Painlevé III asymptotics of Hankel determinants for a perturbed Jacobi weight