Spectrum of random perturbations of Toeplitz matrices with finite symbols

Abstract

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.

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

“…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].…”

The Limit Empirical Spectral Distribution of Gaussian Monic Complex Matrix Polynomials

“…It is immediate by Definitions 2.3 and 2.4 that the ESD of a random matrix polynomial coincides with the ESD of its (random) companion matrix (3). Indeed, in this paper we will strongly rely on its equivalence.…”

“…In Sect. 3, we obtain our first main result: the almost sure limit, for n → ∞, of the empirical spectral distribution of a random n × n monic complex Gaussian matrix polynomial of degree k. In Sect. 4, our second main result is discussed: the almost sure limit, for k → ∞, of the empirical spectral distribution of a random n × n monic complex Gaussian matrix polynomial of degree k. In Sect.…”

“…where each eigenvalue has multiplicity n. It is thus easy to see that the almost sure limit ESD of B is the uniform (singular) probability measure on the unit circle 1 U . The problem can be seen through the lens of the theory of perturbations for Toeplitz matrices and sequences (see, for example, [3,22]), but since the perturbation E 1 Ĉ T is a rank n correction to the Teoplitz matrix B, this case does not fall in the classical settings, where it is required that the random perturbation is not singular with high probability. Since the rank of the perturbation is small when compared with the growing size kn of the matrices, we may anyway expect that the ESD of M also converge almost surely to the same distribution 1 U .…”

Gaussian Regularization of the Pseudospectrum and Davies’ Conjecture

“…In the noncentered case, Davies and Hager [15] showed that if A is a Jordan block and h n for some appropriate , then almost all of the eigenvalues of A g G n lie near a circle of radius 1=n with probability 1 o n .1/. Basak, Paquette, and Zeitouni [4,5] showed that for a sequence of banded Toeplitz matrices A n with a finite symbol, the spectral measures of A n gn G n converge weakly in probability, as n 3 I, to a predictable density determined by the symbol. Both of the above results were recently and substantially improved by Sjöstrand and Vogel [37,38], who proved that for any Toeplitz A, almost all of the eigenvalues of A g n G n are close to the symbol curve of A with exponentially good probability in n. Note that none of the results mentioned in this paragraph explicitly discuss the .…”