Asymptotic eigenvalue distribution of large Toeplitz matrices

Details Asymptotic eigenvalue distribution of large Toeplitz matrices Asymptotic eigenvalue distribution of large Toeplitz matrices

2007-08-23

arxiv.org/abs/0708.3124v1

Physics

Mathematical Physics

Scientific paper

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 $\det(\zeta-T_n(a))$, 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.

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