Outliers of random perturbations of Toeplitz matrices with finite symbols

Abstract

Consider an $N\times N$ Toeplitz matrix $T_N$ with symbol $a(\lambda ):={\sum }_{\ell =-d_2}^{d_1}{a}_{\ell }{\lambda }^{\ell }$, perturbed by an additive noise matrix $N^{-\gamma }{E}_N$, where the entries of $E_N$ are centered i.i.d.~random variables of unit variance and $\gamma >1/2$. It is known that the empirical measure of eigenvalues of the perturbed matrix converges weakly, as $N\to \infty$, to the law of $a(U)$, where $U$ is distributed uniformly on ${\mathbb{S}}^1$. In this paper, we consider the outliers, i.e. eigenvalues that are at a positive ($N$-independent) distance from $a({\mathbb{S}}^1)$. We prove that there are no outliers outside $\mathrm{spec}T(a)$, the spectrum of the limiting Toeplitz operator, with probability approaching one, as $N\to \infty$. {In contrast,} in $\mathrm{spec}T(a)\setminus a({\mathbb{S}}^1)$ 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 $P_{z,a}(\lambda ):=(a(\lambda )-z){\lambda }^{d_2}=0$ and semi-standard Young Tableaux with shapes determined by the number of roots of $P_{z,a}(\lambda )=0$ that are greater than one in moduli.

39 pages, 1 figure; assumption on the Lévy concentration function on the entries relaxed, and the proof of the main theorem simplified