Outliers of random perturbations of Toeplitz matrices with finite symbols

Abstract

Consider an $N\times N$ Toeplitz matrix $T_N$ with symbol ${{\varvec{a}} }(\lambda ) := \sum _{\ell =-d_2}^{d_1} a_\ell \lambda ^\ell$, 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 $N\rightarrow \infty$, to the law of ${{\varvec{a}}}(U)$, 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 ${{\varvec{a}}}({\mathbb {S}}^1)$. We prove that there are no outliers outside $\mathrm{spec} \, T({{\varvec{a}}})$, the spectrum of the limiting Toeplitz operator, with probability approaching one, as $N \rightarrow \infty$. In contrast, in $\mathrm{spec}\, T({{\varvec{a}}}){\setminus } {{\varvec{a}}}({{\mathbb {S}}}^1)$ 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 $P_{z, {{\varvec{a}}}}(\lambda ):= ({{\varvec{a}}}(\lambda ) -z)\lambda ^{d_2}$ and semi-standard Young Tableaux with shapes determined by the number of roots of $P_{z,{{\varvec{a}}}}(\lambda )=0$ that are greater than one in moduli.

Appendix A: The spectral radius of \(E_N\)

In this short section we show that the decomposition (2.2) used in the proofs of Theorems 1.1 and 1.11 can be adapted to prove the following.

Proposition A.1

Let $\{E_N\}_{N \in {\mathbb {N}}}$ be a sequence of $N \times N$ random matrices with independent complex-valued entries of mean zero and unit variance. Denote $\varrho _N$ to be the spectral radius of $N^{-1/2} E_N$, i.e. the maximum modulus eigenvalue of $N^{-1/2} E_N$. Then the sequence $\{\varrho _N\}_{N \in {\mathbb {N}}}$ is tight.

We remark that Proposition A.1 seems to be contained in Theorem 1.1. However, formally the latter cannot be applied since it would require one to take ${\varvec{a}} \equiv 0$, while throughout the paper (and in particular, in the proof of Theorem 1.1), we assume that ${\varvec{a}}$ is a nontrivial Laurent polynomial.

If the entries of $E_N$ are i.i.d. having a finite $(2+\delta )$-th moment and possessing a symmetric law then it is known that $\varrho _N\rightarrow 1$ in probability, see [6], while the operator norm of $N^{-1/2} E_N$ blows up as soon as the fourth moment of the entries is infinite. It is conjectured in [6] that in the critical case of finiteness of second moments, the convergence in probability to one still holds. Proposition A.1 is a weak form of the conjecture with elementary proof.

Proof

Set $\Delta _N= N^{-1/2} E_N$. We decompose

$$\begin{aligned} \det (\Delta _N-zI_N)=(-z)^N+\sum _{k=1}^{N} (-z)^{N-k}P_k \end{aligned}$$

where

$$\begin{aligned} P_k:=\sum _{\begin{array}{c} X\subset [N]\\ |X|=k \end{array}} N^{- k/2} \det (E_N[X; X]), \end{aligned}$$

(A.1)

compare with (2.1). Note that ${\text {Var}}(P_k)\le 1$, while ${\mathbb {E}}P_k=0$. Therefore, for a fixed constant ${\bar{C}}$, we have that ${\mathbb {P}}(|P_k|>{\bar{C}}^k)\le {\bar{C}}^{-2k}$. So, setting ${\mathcal {A}}_0=\cup _{k=1}^\infty \{ |P_k|>{\bar{C}}^k\}$, it yields that

$$\begin{aligned} {\mathbb {P}}({\mathcal {A}}_0)\le \frac{2}{ {\bar{C}}^2}. \end{aligned}$$

Note that on ${\mathcal {A}}_0^c$ we have that for z with $|z|>4 {\bar{C}}$,

$$\begin{aligned} \frac{|z^N|}{|\sum _{k=1}^{N} (-z)^{N-k}P_k|}\ge \frac{1}{\sum _{k=1}^N 4^{-k}}\ge 3. \end{aligned}$$

This in particular implies that there can be no zero of $\det (\Delta _N-zI_N)$ with modulus larger than $4{\bar{C}}$. Thus the claim follows. $\square$