On the asymptotics of a Toeplitz determinant with singularities

Summary

We provide an alternative proof of the classical single-term asymptotics for Toeplitz determinants whose symbols possess Fisher–Hartwig singularities. We also relax the smoothness conditions on the regular part of the symbols and obtain an estimate for the error term in the asymptotics. Our proof is based on the Riemann–Hilbert analysis of the related systems of orthogonal polynomials and on differential identities for Toeplitz determinants. The result discussed in this paper is crucial for the proof of the asymptotics in the general case of Fisher–Hartwig's singularities and extensions to Hankel and Toeplitz+Hankel determinants.

1. Introduction

Let f(z) be a complex-valued function integrable over the unit circle. Denote its Fourier coefficients

We are interested in the n-dimensional Toeplitz determinant with symbol f( z),

where f (e^{i θ}) has a fixed number of Fisher–Hartwig singularities [Fisher and Hartwig 1968; Lenard 1964; 1972], i.e., f has the following form on the unit circle:

for some m = 0; 1; : : : , where

and V(e_{i θ}) is a sufficiently smooth function on the unit circle (see below). The condition on the _j insures integrability. Note that a single Fisher–Hartwig singularity at z j consists of a root-type singularity

and a jump e^{iπ β}We assume that z_j, j =1,...,m, are genuine singular points, i.e., either j or βj=0. However, we always include z₀=1 explicitly in (1-2), even when _0 = β=0 : this convention was adopted in [Deift et al. 2011] in order to facilitate the application of our Toeplitz methods to Hankel determinants. Note that g_βo(z)= e(^-iπβ₀). Observe that for each j≠0, z^β_jgβj(z) is continuous at z=1, and so for each j each “beta” singularity produces a jump only at the point z_j. The factors z_j^-βθ are singled out to simplify comparisons with the existing literature. Indeed, (1-2) with the notation b(θ) = e^V(eiθ) is exactly the symbol considered in [Fisher and Hartwig 1968; Basor 1978; 1979; Böttcher and Silbermann 1981; 1985; 1986; Ehrhardt and Silbermann 1997; Ehrhardt 2001; Widom 1973]. However, we write the symbol in a form with _zΣ^m_j=0β_j factored out. The representation (1-2) is more natural for our analysis.