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

Abstract

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 .

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].…”

“…Theorems 1.4 and 1.5 are closely related to a number of techniques used in the random matrix theory literature to study non-Hermitian matrices, including those found in [5,7,24,65]. The authors are not aware of any works where these results are stated in the deterministic forms given above.…”

Deterministic equivalence for noisy perturbations

“…evaluating logarithmic potentials amounts to computing determinants. In their study of the spectrum of small, noisy perturbations of non-normal matrices, the authors of [1] have identified a certain deterministic equivalent result, which we now present.…”

“…The proof in [1] uses in an essential way the unitary invariance of G N , and probabilistic arguments. However, it does not directly extend to other noise models, not even to the case where G N is a matrix consisting of independent real standard Gaussian variables.…”

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 .…”