A Status Report on the Asymptotic Behavior of Toeplitz Determinants with Fisher-Hartwig Singularities

Thinning and conditioning of the circular unitary ensemble

“…For the eigenvalue correlation kernel K n , one expects from the analysis in [14] to have the scaling limit 19) for any u, v ∈ R and θ ∈ [0, 2π] \ {L, 2π − L}, and where c = . Near e ±iL , the asymptotics for φ n are described in terms of confluent hypergeometric functions, and instead of the sine kernel, one expects a limiting kernel built out of confluent hypergeometric functions.…”

“…If L ∈ (0, π) and s ∈ (0, 1) are independent of n, large n asymptotics for D n (s, L) are known, and we have [14,19] …”

An Introduction to Integrable Techniques for One-Dimensional Quantum Systems

“…Virtually all static correlation functions of the model can be expressed as determinants of matrices with a special structure, known as Toeplitz matrices [17]. The asymptotic behavior of Toeplitz determinants can be studied using fairly sophisticated mathematical techniques or just by relying on known theorems, such as the Szegö Theorem, the Fisher-Hartwig conjecture, Widom's theorem and so on ... [18,19] The phase diagram of this model is parametrized by the anisotropy parameter γ capturing the relative strength of interaction in the x and y components and by the external magnetic field h, directed along the transverse z-axis. We take these parameters to be dimensionless and from now on we set the energy-scale defining parameter as J = −1 (that is, we will consider an easy-plane ferromagnet).…”

Norms of Toeplitz Matrices with Fisher–Hartwig Symbols

“…We will also establish results for symbols with more than one Fisher-Hartwig singularity. For the exciting story behind Toeplitz matrices with Fisher-Hartwig symbols and their determinants we refer to the books [4], [5] and the papers [1], [9]. For general Toeplitz matrices, the asymptotic distribution of the singular values and the asymptotics of the extreme singular values has been studied by many authors, and we allow us to abstain from giving an ample list of references here.…”