Universality of the ESD for a fixed matrix plus small random noise: A stability approach

Abstract

We study the empirical spectral distribution (ESD) in the limit where n → ∞ of a fixed n by n matrix M n plus small random noise of the form f (n)X n , where X n has iid mean 0, variance 1/n entries and f (n) → 0. It is known for certain M n , in the case where X n is iid complex Gaussian, that the limiting distribution of the ESD of M n + f (n)X n can be dramatically different from that for M n . We prove a general universality result showing, with some conditions on M n and f (n), that the limiting distribution of the ESD does not depend on the type of distribution used for the random entries of X n . We use the universality result to exactly compute the limiting ESD for two families where it was not previously known. The proof of the main result incorporates the Tao-Vu replacement principle and a version of the Lindeberg replacement strategy, along with the newly-defined notion of stability of sets of rows of a matrix.

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

“…Previous results on typical perturbations and strategy. Our work is an attempt to address the same issue from a "typical" perturbation point of view, and in this sense continues the line of research initiated in [10,27] and [6], which we now describe.…”

“…They further show that their assumptions are satisfied when M N = J. The paper [27] shows that this result is stable in the sense that replacing G N by a matrix with i.i.d. entries satisfying mild assumptions does not change the result.…”

“…We believe that the results should carry over in a rather straightforward way to the case of real Ginibre matrices, and with a significant effort to the i.i.d. setup, in the same spirit as [27].…”

Regularization of Non-Normal Matrices by Gaussian Noise

“…In particular it would be of interest to remove the restriction on (N ). As in [Woo14], the Gaussian assumption on the noise probably could also be weakened, though this requires a better understanding of how small det(G + C) can be, where G is a matrix of i.i.d. elements and C is some arbitrary matrix (in particular, without a priori estimates on the norm of C); such control is not currently available for small singular values of G + C without putting some a-priori conditions on C, see e.g.…”

Approximation to stable law by the Lindeberg principle

“…[8,7] obtained a general scaling of disordered system by expanding polynomial chaos. Besides a lot of applications in random matrices [4,6,26,41,29,42,1,5], Lindeberg principle has also been applied to other research 1 areas such as high dimensional regressions [14,15,20], time series [21,41,32,31], bootstrap [35,15], statistical learning [27,44] and so on.…”