Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities
type: Item
instance of: scholarly article
title: Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities
zbMATH Open document ID: 1232.15006
published in: Annals of Mathematics. Second Series
publication date: 2011-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/0905.0443
review text: The asymptotics when \(n\rightarrow \infty\) for \(n\)-dimensional Toeplitz, Hankel and Toeplitz+Hankel determinants whose symbols possess Fisher-Hartwig singularities are studied. A symbol with Fisher-Hartwig singularities is a function \(f(z)\) on the unit circle such that \(f(z) = e^{V(z)} z^{\sum_{j=0}^m \beta_j} \prod_{j=0}^m |z-z_j|^{2\alpha_j}g_{z_j,\beta_j}(z) z_j^{-\beta_j}\), \(z=e^{i\theta}\), \(\theta\in[0,2\pi]\), \(z_j=e^{i\theta_j}\), \(j=0,\dots,m\), \(0=\theta_0< \theta_1<\dots <\theta_m<2\pi\), \(g_{z_j,\beta_j}(z)=e^{i\pi\beta_j}\) if \(0\leq \mathrm{arg} z\leq \theta _j\), and \(e^{-i\pi\beta_j}\) if \(\theta_j\leq arg z\leq 2\pi\), \(\mathrm{Re}\alpha_j>-\frac{1}{2}\), \(\beta_j\in\mathbb{C}\), \(V(z)\) is a sufficiently smooth function on the unit circle. Let \(f_j=\frac{1}{2\pi}\int_0^{2\pi}f(e^{i\theta})e^{-ij\theta}d\theta\) be the Fourier coefficients of \(f\). Then the Toeplitz determinant with the symbol \(f\) is defined as \(D_n(f)=\det(f_{j-k})_{j,k=0}^{n-1}\), the Hankel determinant with the symbol \(f\) is defined as \(D_n(f)=\det(f_{j+k+1})_{j,k=0}^{n-1}\), and determinant of types \(D_n(f)=\det(f_{j-k}+f_{j+k})_{j,k=0}^{n-1}\), \(D_n(f)=\det(f_{j-k}-f_{j+k+2})_{j,k=0}^{n-1}\), \(D_n(f)=\det(f_{j-k}\pm f_{j+k+1})_{j,k=0}^{n-1}\) are called the Toeplitz+Hankel determinant. The determinants of such types corresponding to symbols with Fisher-Hartwig singularities have applications in statistical mechanics [\textit{M. E. Fisher} and \textit{R. E. Hartwig}, ``Toeplitz determinants: some applications, theorems, and conjectures'', Adv. Chem. Phys. 15, 333--353 (1969)], in random matrix theory [\textit{M. L. Mehta}, Random matrices. Rev. and enlarged 2. ed. Boston, MA: Academic Press, Inc. (1991; Zbl 0780.60014)], and even in number theory [\textit{É. Brézin} (ed.) et al., Application of random matrices in physics. NATO Science Series II: Mathematics, Physics and Chemistry 221. Dordrecht: Springer. (2006; Zbl 1120.82001)]. The first formula …
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 15B52
zbMATH DE Number: 5960727
zbMATH Keywords: Toeplitz determinant
zbMATH Keywords: Hankel determinant
zbMATH Keywords: Toeplitz+Hankel determinant
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: Fisher-Hartwig singularities
author: Percy A. Deift
author: A. R. Its
author: I. V. Krasovsky
reviewed by: Dmitry Artamonov
MaRDI profile type: Publication
OpenAlex ID: W2963471558
arXiv ID: 0905.0443
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DOI: 10.4007/ANNALS.2011.174.2.12
description: scientific article