Asymptotic eigenvalue distribution of large Toeplitz matrices

Lee, Seung-Yeop; Dai, Hui; Bettelheim, Eldad

Asymptotic eigenvalue distribution of large Toeplitz matrices

Authors

Lee, Seung-Yeop

Dai, Hui

Bettelheim, Eldad

Description

We study the asymptotic eigenvalue distribution of Toeplitz matrices generated by a singular symbol. It has been conjectured by Widom that, for a generic symbol, the eigenvalues converge to the image of the symbol. In this paper we ask how the eigenvalues converge to the image. For a given Toeplitz matrix $T_n(a)$ of size $n$, we take the standard approach of looking at $\det(ζ-T_n(a))$, of which the asymptotic information is given by the Fisher-Hartwig theorem. For a symbol with single jump, we obtain the distribution of eigenvalues as an expansion involving $1/n$ and $\log n/n$. To demonstrate the validity of our result we compare our result against the numerics using a pure Fisher-Hartwig symbol.