2019/05/24 by Basak, Anirban, Zeitouni, Ofer · 1 citation
#FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1905.10244
Consider an N× N Toeplitz matrix TN with symbol a (λ) := ∑ℓ=-d2d1 a_ℓ λ^ℓ, perturbed by an additive noise matrix N-γ EN, where the entries of EN 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 \mathbbS1. In this paper, we consider the outliers, i.e. eigenvalues that are at a positive (N-independent) distance from a(\mathbbS1). We prove that there are no outliers outside \rm spec T(a), the spectrum of the limiting Toeplitz operator, with probability approaching one, as N → ∞. In contrast, in \rm spec T(a)∖ a(\mathbb S1) 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 EN. The coefficients in the linear combination depend on the roots of the polynomial P_z, a(λ):= (a(λ) -z)λd2=0 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.