Toeplitz determinants with perturbations in the corners

TL;DR

This paper derives exact and asymptotic formulas for Toeplitz determinants with corner perturbations, including cases with Fisher-Hartwig singularities, advancing understanding of their asymptotic behavior.

Contribution

It provides new formulas for Toeplitz determinants with corner perturbations, especially addressing Fisher-Hartwig singularities, which were not covered by standard perturbation theory.

Findings

Established formulas for Toeplitz determinants with corner perturbations.

Derived asymptotic expressions for matrices with Fisher-Hartwig singularities.

Extended the theory to Hermitian matrices induced by Fisher-Hartwig symbols.

Abstract

The paper is devoted to exact and asymptotic formulas for the determinants of Toeplitz matrices with perturbations by blocks of fixed size in the four corners. If the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity, then standard perturbation theory yields asymptotic expressions for the perturbed determinants. This premise is not satisfied for matrices generated by so-called Fisher-Hartwig symbols. In that case we establish formulas for pure single Fisher-Hartwig singularities and for Hermitian matrices induced by general Fisher-Hartwig symbols.