Large bidiagonal matrices and random perturbations

Sjöstrand, Johannes; Vogel, Martin

doi:10.4171/jst/150

Cited by 14 publications

(8 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The choice of γ in each of the three cases given in Corollary 1.8 is so that (1.11) is satisfied. Theorem 1.7 and its corollary are similar to several recent works concerning fixed matrices perturbed by random matrices [7,8,10,29,60,61,62,73]. The case when E contains independent Gaussian entires was investigated in [7,29,60,61,62].…”

Section: 14)supporting

confidence: 59%

“…Theorem 1.7 and its corollary are similar to several recent works concerning fixed matrices perturbed by random matrices [7,8,10,29,60,61,62,73]. The case when E contains independent Gaussian entires was investigated in [7,29,60,61,62]. Since Theorem 1.7 applies to a large class of perturbations E, it is more closely related to the results in [8,73].…”

Section: 14)mentioning

confidence: 52%

See 1 more Smart Citation

Quantitative results for non-normal matrices subject to random and deterministic perturbations

O’Rourke¹,

Wood²

2021

Preprint

View full text Add to dashboard Cite

We consider the eigenvalue distribution of a fixed matrix subject to a small perturbation. In particular, we prove quantitative comparison results which show that the logarithmic potential is stable under perturbations with small norm or low rank, provided the smallest and largest singular values are not too extreme. We also establish a quantitative version of the Tao-Vu replacement principle. As an application of our results, we study the spectral distribution of banded Toeplitz matrices subject to small random perturbations, including the case when the bandwidth grows with the dimension. For this model, we obtain a rate of convergence in Wasserstein distance of the empirical spectral measure to its limiting distribution. We give a number of examples of our methods including random multiplicative perturbations and large classes of deterministic perturbations that behave similar to random perturbations.

show abstract

Section: 14)supporting

confidence: 59%

Section: 14)mentioning

confidence: 52%

Quantitative results for non-normal matrices subject to random and deterministic perturbations

O’Rourke¹,

Wood²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…We note that other approaches to the study of perturbations of non-normal operators exist. In particular, Sjöstrand and Vogel [17], [18] identify the limit of the empirical value of a random perturbation of a banded Toeplitz matrix with two non-zero diagonals, one above and one below the main diagonal. Their methods, which are quite different from ours, are limited to γ > 5/2, and yield more quantitative estimates on the empirical measure and its outliers.…”

Section: 2mentioning

confidence: 99%

Regularization of Non-Normal Matrices by Gaussian Noise—the Banded Toeplitz and Twisted Toeplitz Cases

Basak

Paquette

Zeitouni

2019

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let M N be a deterministic N × N matrix, and let G N be a complex Ginibre matrix. We consider the matrix M N = M N + N −γ G N , where γ > 1/2. With L N the empirical measure of eigenvalues of M N , we provide a general deterministic equivalence theorem that ties L N to the singular values of z − M N , with z ∈ C. We then compute the limit of L N when M N is an upper triangular Toeplitz matrix of finite symbol: if M N = d i=0 a i J i where d is fixed, a i ∈ C are deterministic scalars and J is the nilpotent matrix J(i, j) = 1 j=i+1 , then L N converges, as N → ∞, to the law of d i=0 a i U i where U is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when d = 1, also of i.i.d. entries on the diagonals in M N .

show abstract

“…Hence, in recent times there have been growing interests in studying the spectrum of non-normal operators and matrices under small typical perturbations. See the references in [16,Section 1]. We also refer to [2, Section 1.3] for a discussion about the relation between the pseudospectrum and the spectrum under typical perturbation, and an extensive reference list.…”

Section: Introductionmentioning

confidence: 99%

Spectrum of random perturbations of Toeplitz matrices with finite symbols

Basak

Paquette

Zeitouni

2020

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

Large bidiagonal matrices and random perturbations

Cited by 14 publications

References 15 publications

Quantitative results for non-normal matrices subject to random and deterministic perturbations

Quantitative results for non-normal matrices subject to random and deterministic perturbations

Regularization of Non-Normal Matrices by Gaussian Noise—the Banded Toeplitz and Twisted Toeplitz Cases

Spectrum of random perturbations of Toeplitz matrices with finite symbols

Contact Info

Product

Resources

About