Asymptotics for Toeplitz determinants: perturbation of symbols with a gap

Charlier, Christophe;Claeys, Tom
(2015) Journal of Mathematical Physics — Vol. 56, n° 2, p. 22705 (2015)

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We study 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.
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Charlier, C., & Claeys, T. (2015). Asymptotics for Toeplitz determinants: perturbation of symbols with a gap. Journal of Mathematical Physics, 56(2), 22705. https://doi.org/10.1063/1.4908105 (Original work published 2015)