Outliers of random perturbations of Toeplitz matrices with finite symbols.

Consider an N × N Toeplitz matrix T N with symbol a (λ) : = ∑ ℓ = - d 2 d 1 a ℓ λ ℓ , perturbed by an additive noise matrix N - γ E N , where the entries of E N are centered i.i.d. 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 P z , a (λ) : = (a (λ) - z) λ d 2 and semi-standard Young Tableaux with shapes determined by the number of roots of P z , a (λ) = 0 that are greater than one in moduli. [ABSTRACT FROM AUTHOR]