A Note on Wiener-Hopf Determinants and the Borodin-Okounkov Identity

Abstract

The continuous analogue of a Toeplitz determinant identity for Wiener-Hopf operators is proved. An example which arises from random matrix theory is studied and an error term for the asymptotics of the determinant is computed. Wiener-Hopf DeterminantsRecently, a beautiful identity due to Borodin and Okounkov was proved for Toeplitz determinants which shows how one can write a Toeplitz determinant as a Fredholm determinant. In this note we generalize this to the Wiener-Hopf case. The proof in the Wiener-Hopf case follows identically with the second one given in [1]. We include it here for completeness sake and because the nature of the identity is slightly different in the continuous verses discrete convolution setting.In the Wiener-Hopf case we begin with a Fredholm determinant on a finite interval and then show how this can be written as a Fredholm determinant of an operator defined on L 2 of a half-line. The point is the second operator has a very "small" kernel and thus higher order approximations (as a function of the length of the finite interval) can be found.We now state the analogue of the identity and then apply it to a particular case to show how error estimates can be computed. In the future we hope to refine the estimates given here, apply this identity to other important examples, and also extend it to other operators.We consider the Fredholm determinant of the finite Wiener-Hopf operator det(I − K [0,α] ), (1.1)

Wiener-Hopf Determinants with Fisher-Hartwig Symbols

“…where σ 0 is a symbol for which (2) holds. For an attempted proof, it seems reasonable to try to do what was done in the Toeplitz case, that is, devise the proper localization techniques and then try to evaluate the determinants in the case of the pure singularity.…”

“…This will be proved by finding exact formulas for the Toeplitz and Wiener-Hopf determinants and regularized determinants and then showing the above quotients are asymptotically equal when R ∼ 2n. These formulas are expressed in terms of Fredholm determinants of operators acting on L 2 (0, 1) and are obtained by using an identity of Borodin and Okounkov [5] for Toeplitz determinants with regular symbol and its Wiener-Hopf analogue [2]. What we do is simply stated: in both cases we introduce a parameter to regularize the symbol, apply the identity, and then take the limit.…”

Analysis of Toeplitz Operators

“…The Wiener-Hopf analogue of the Borodin-Okounkov formula was established by Basor and Chen [26] and Basor and Widom [43].…”

On Determinant Expansions for Hankel Operators

“…The Wiener-Hopf operator on L 2 (0, ∞) is W (F ) : f (x) → ∞ 0 K(x − y)f (y) dy. The Wiener-Hopf factorization 1 − F (ξ) = ψ − (iξ)ψ + (iξ) (1.4) was considered by Basor and Chen [2], who obtained various identities for determinants of related Hankel operators on L 2 (0, ∞). The following integral plays a central role in their analysis…”