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_ℓλ ^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.