Asymptotics for Toeplitz determinants: perturbation of symbols with a gap. (English) Zbl 1310.15051
Summary: 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 H. Widom [Indiana Univ. Math. J. 21, 277–283 (1971; Zbl 0213.34903)], 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.{
©2015 American Institute of Physics}
©2015 American Institute of Physics}
MSC:
| 15B05 | Toeplitz, Cauchy, and related matrices |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
Keywords:
determinants; Toeplitz matrices; asymptotic expansion; circular unitary ensemble; Fredholm determinants; sine kernelCitations:
Zbl 0213.34903References:
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