Asymptotics for Toeplitz determinants: perturbation of symbols with a gap

Abstract

We study the determinants of Toeplitz matrices as the size of the matrices tends to infinity, in the particular case where the symbol has two jump discontinuities and tends to zero on an arc of the unit circle at a sufficiently fast rate. We generalize an asymptotic expansion by Widom [22], which was known for symbols supported on an arc. We highlight applications of our results in the Circular Unitary Ensemble and in the study of Fredholm determinants associated to the sine kernel.

28 pages, 5 figures

References in corpus (2)

• On the asymptotics of a Toeplitz determinant with singularities
• Asymptotics for Toeplitz determinants on a circular arc

Cited by in corpus (3)

• Asymptotic expansion of Toeplitz determinants of an indicator function with discrete rotational symmetry and powers of random unitary matrices
• Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation
• Asymptotics of the partition function of the perturbed Gross-Witten-Wadia unitary matrix model