6533b856fe1ef96bd12b2680
RESEARCH PRODUCT
Toeplitz band matrices with small random perturbations
Martin VogelJohannes Sjöstrandsubject
Pure mathematicsSpectral theoryGeneral Mathematics010103 numerical & computational mathematics01 natural sciencesMathematics - Spectral TheoryMathematics - Analysis of PDEsFOS: MathematicsAsymptotic formula0101 mathematicsSpectral Theory (math.SP)Eigenvalues and eigenvectorsMathematics010102 general mathematicsProbability (math.PR)Toeplitz matrixComplex normal distribution[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]Weyl lawRandom perturbationsRandom matrixComplex planeSpectral theoryMathematics - ProbabilityNon-self-adjoint operators[MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]Analysis of PDEs (math.AP)description
We study the spectra of $N\times N$ Toeplitz band matrices perturbed by small complex Gaussian random matrices, in the regime $N\gg 1$. We prove a probabilistic Weyl law, which provides an precise asymptotic formula for the number of eigenvalues in certain domains, which may depend on $N$, with probability sub-exponentially (in $N$) close to $1$. We show that most eigenvalues of the perturbed Toeplitz matrix are at a distance of at most $\mathcal{O}(N^{-1+\varepsilon})$, for all $\varepsilon >0$, to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2021-02-01 |