Estelle L. Basor

The Fisher-Hartwig conjecture and generalizations

We discuss the status of the Fisher-Hartwig conjecture concerning the asymptotic expansion of a class of Toeplitz determinants with singular generating functions. A counterexample is given for a nonrational generating function; and we formulate a generalized Fisher-Hartwig conjecture.

On a Toeplitz determinant identity of Borodin and Okounkov

In this note we give two other proofs of an identity of A. Borodin and A. Okounkov which expresses a Toeplitz determinant in terms of the Fredholm determinant of a product of two Hankel operators. The second of these proofs yields a generalization of the identity to the case of block Toeplitz determinants.

The Fisher-Hartwig conjecture and Toeplitz eigenvalues

Abstract The conjecture of Fisher and Hartwig, published in 1968, describes the asymptotic expansion of Toeplitz determinants with singular generating functions. For more than twenty years progress was made in extending the validity of the conjecture, but recent computer experiments led to counterexamples that show the limits of the original conjecture and pointed the way to a revised conjecture. This paper describes the history of the problem, several numerical examples, and the revised conjecture.

Toeplitz and Wiener-Hopf determinants with piecewise continuous symbols

Abstract The asymptotics for determinants of Toeplitz and Wiener-Hopf operators with piecewise continuous symbols are obtained in this paper. If Wα(σ) is the Wiener-Hopf operator defined on L2(0, α) with piecewise continuous symbol σ having a finite number of discontinuities at ξr, then under appropriate conditions it is shown that det Wα(σ) ~ G(σ)α αΣλr2K(σ), where G(σ) = exp(log σ) ( 0), λ r = ( 1 2π ) log [ σ(ξ r +) σ(ξ r −) ] and K(σ) is a completely determined constant. An analogous result is obtained for Toeplitz operators. The main point of the paper is to obtain a result in the Wiener-Hopf case since the Toeplitz case had been treated earlier. In the Toeplitz case it was discovered that one could obtain asymptotics fairly easily for symbols with several singularities if, for each singularity one could find a single example of a symbol with a singularity of that kind whose associated asymptotics were known. Fortunately in the Toeplitz case such asymptotics were known. The difficulty in the Wiener-Hopf case is that there was not a single singular case where the determinant was explicitly known. This problem was overcome by using the fact that Wiener-Hopf determinants when discretized become Toeplitz determinants whose entries depend on the size of the matrix. No theorem on Toeplitz matrices can be applied directly but these theorems are modified to obtain the desired results.

Distribution Functions for Random Variables for Ensembles of Positive Hermitian Matrices

Abstract:Distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive Hermitian matrices. The inverse Fourier transform of the distribution is shown to be a Fredholm determinant of a certain operator that is an analogue of a Wiener-Hopf operator. The asymptotic formula shows that, up to the terms of order o(1), the distributions are Gaussian.

Asymptotic Formulas for Determinants of a Sum of Finite Toeplitz and Hankel Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.

Perturbed Hankel determinants

In this paper, we compute, for largen, the determinant of a class of n×n Hankel matrices, which arise from a smooth perturbation of the Jacobi weight. For this purpose, we employ the same idea used in previous papers, where the unknown determinant Dn[wα,βh] is compared with the known determinant Dn[wα,β]. Here wα,β is the Jacobi weight and wα,βh, where h = h(x), x ∈ [−1, 1], is strictly positive and real analytic, is the smooth perturbation on the Jacobi weight wα,β (x) := (1 − x) α ( 1+ x) β . Applying a previously known formula on the distribution function of linear statistics, we compute the large-n asymptotics of Dn[wα,βh] and supply a missing constant of the expansion.

ASYMPTOTICS OF A TAU-FUNCTION AND TOEPLITZ DETERMINANTS WITH SINGULAR GENERATING FUNCTIONS

We compute the short distance asymptotics of a tau-fucntion appearing in the work of Sato, Miwa, and Jimbo on holonomic quantum fields, We show that these asymptotics are determined by the Widom operator. This same operator is fundamental in the asymptotics of Toeplitz determinants with singular generating fucntions.

Determinants of Airy Operators and Applications to Random Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener–Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at “the edge of the spectrum.”

Some identities for determinants of structured matrices

Abstract In this paper we establish several relations between the determinants of the following structured matrices: Hankel matrices, symmetric Toeplitz + Hankel matrices and Toeplitz matrices. Using known results for the asymptotic behavior of Toeplitz determinants, these identities are used in order to obtain Fisher–Hartwig type results on the asymptotics of certain skew-symmetric Toeplitz determinants and certain Hankel determinants.