Toeplitz Matrices and Toeplitz Determinants under the Impetus of the Ising Model: Some History and Some Recent Results

Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

“…[20] for more on the history of the matter), the evaluation of the constant terms in the asymptotics of different correlation and distribution functions of random matrix theory and of the theory of solvable statistical mechanics models has always been a great challenge in the field 1 .…”

Random Matrices with Merging Singularities and the Painlevé V Equation

“…The singularities e ±t approach the unit circle as t → 0, and form a Fisher-Hartwig type singularity if t = 0. Asymptotics for Toeplitz determinants with weight functions of this form were also obtained in the context of the 2d Ising model [32], see [9] for a review of Toeplitz determinants and the Ising model. Theorem 1.1 in [4] states that…”

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).…”