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Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities

P. Deift, A. Its, I. Krasovsky

Abstract

We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general non-degenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.

Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities

Abstract

We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general non-degenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.

Paper Structure

This paper contains 12 sections, 14 theorems, 233 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Orthogonal polynomials on the unit circle. Toeplitz and Hankel determinants.
  3. Riemann-Hilbert problem
  4. Asymptotic analysis of the Riemann-Hilbert problem
  5. Parametrix outside the points $z_j$
  6. Parametrix at $z_j$.
  7. R-RHP
  8. Orthogonal polynomials. Proof of Theorem \ref{['poly']}
  9. Toeplitz determinants. Proof of Theorem \ref{['BT']}
  10. The case of analytic $V(z)$
  11. Extension to smooth $V(z)$
  12. Hankel determinants. Proof of Theorem \ref{['asHankel']}

Key Result

Theorem 1.1

(Ehrhardt Ehr). Let $f(e^{i\theta})$ be defined in (fFH), $V(z)$ be $C^\infty$ on the unit circle, $|||\beta|||<1$, $\Re\alpha_j>-1/2$, and $\alpha_j\pm\beta_j\neq -1,-2,\dots$ for $j,k=0,1,\dots,m$. Then as $n\to\infty$, where $G(x)$ is Barnes' $G$-function. The double product over $j<k$ is set to $1$ if $m=0$.

Figures (3)

  • Figure 1: Contour for the $S$-Riemann-Hilbert problem ($m=2$).
  • Figure 2: The auxiliary contour for the parametrix at $z_j$.
  • Figure 3: Contour $\Gamma$ for the $R$ and $\widetilde{R}$ Riemann-Hilbert problems ($m=2$).

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 35 more