Interior Eigenvalue Density of Jordan Matrices with Random Perturbations

Sjöstrand, Johannes; Vogel, Martin

doi:10.1007/978-3-319-52471-9_24

Cited by 4 publications

(8 citation statements)

References 23 publications

(36 reference statements)

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“…Our Theorem 1.11, when applied to non-triangular tridiagonal Toeplitz matrix, again shows that under complex Gaussian perturbation the limiting random fields are the zero sets of Gaussian analytic functions, and a computation (which we omit) shows that its covariance kernel is given by K d (·, ·). Thus, Theorem 1.11 again recovers the results of [17].…”

Section: Introductionsupporting

confidence: 75%

“…Sharper results concerning outliers for the Jordan matrix and the non-triangular tridiagonal Toeplitz matrix (under complex Gaussian perturbation), are presented in [17,18]. In both these cases, a sharp O(1) control on the number of outliers in the regions S d with d = 0 is provided.…”

Section: Introductionmentioning

confidence: 99%

“…We now explain how to recover this result from Theorem 1.11: in the case of the Jordan matrix, a(λ) = λ and then λ 1 (z) = z, d = 1 and d 0 = 0. Substituting in Definition 1.5, one finds that Sjöstrand and Vogel in [17] compute ρ d for the non-triangular tridiagonal Toeplitz matrix. Again by the Edelman-Kostlan formula, they identify ρ d with the first intensity measure of the random point process induced by the zero set of some Gaussian analytic function with some covariance kernel K d (·, ·).…”

Section: Introductionmentioning

confidence: 99%

“…Based on [17,18] one may be tempted to predict that for general finitely banded Toeplitz matrices the limiting random field is the zero set of some Gaussian analytic function. Theorem 1.11 shows that, even under complex Gaussian perturbations, the limit may not be the zero set of Gaussian analytic functions, e.g.…”

Section: Introductionmentioning

confidence: 99%

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Spectrum of random perturbations of Toeplitz matrices with finite symbols

Basak

Paquette

Zeitouni

2020

Trans. Amer. Math. Soc.

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Consider an N × N Toeplitz matrix T N with symbol a(λ) := d 1 =−d 2 a λ , perturbed by an additive noise matrix N −γ E N , where the entries of E N are centered i.i.d. complex random variables of unit variance and γ > 1/2. It is known that the empirical measure of eigenvalues of the perturbed matrix converges weakly, as N → ∞, to the law of a(U ), where U is distributed uniformly on S 1 . In this paper, we consider the outliers, i.e. eigenvalues that are at a positive (N -independent) distance from a(S 1 ). We prove that there are no outliers outside spec T (a), the spectrum of the limiting Toeplitz operator, with probability approaching one, as N → ∞. In contrast, in spec T (a) \ a(S 1 ) the process of outliers converges to the point process described by the zero set of certain random analytic functions. The limiting random analytic functions can be expressed as linear combinations of the determinants of finite sub-matrices of an infinite dimensional matrix, whose entries are i.i.d. having the same law as that of E N . The coefficients in the linear combination depend on the roots of the polynomial Pz,a(λ) := (a(λ) − z)λ d 2 = 0 and semi-standard Young Tableaux with shapes determined by the number of roots of Pz,a(λ) = 0 that are greater than one in moduli. We remark that if one is interested in the case where d 1 = 0 but d 2 > 0, one may simply consider, when computing spectra, the transpose of T N or of M N = T N + ∆ N . For this reason, the restriction to d 1 > 0 does not reduce generality.

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Section: Introductionsupporting

confidence: 75%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spectrum of random perturbations of Toeplitz matrices with finite symbols

Basak

Paquette

Zeitouni

2020

Trans. Amer. Math. Soc.

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“…In [15], J. Sjöstrand obtained a probabilistic circular Weyl law for most of the eigenvalues of large Jordan block matrices subject to small random perturbations, and in [17], we obtained a precise asymptotic formula for the average density of the residual eigenvalues in the interior of a circle, where the result of Davies and Hager yielded a logarithmic upper bound on the number of eigenvalues. The leading term is given by the hyperbolic volume form on the unit disk, independent of the dimension N .…”

mentioning

confidence: 99%