Regularization of non-normal matrices by Gaussian noise - the banded Toeplitz and twisted Toeplitz cases

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let $M_N$ be a deterministic $N\times N$ matrix, and let $G_N$ be a complex Ginibre matrix. We consider the matrix ${\mathcal{M}}_N=M_N+N^{-\gamma }{G}_N$, where $\gamma >1\mathrm{/}2$. With $L_N$ the empirical measure of eigenvalues of ${\mathcal{M}}_N$, we provide a general deterministic equivalence theorem that ties $L_N$ to the singular values of $z-M_N$, with $z\in \mathbb{C}$. We then compute the limit of $L_N$ when $M_N$ is an upper triangular Toeplitz matrix of finite symbol: if $M_N={\sum }_{i=0}^{\mathfrak{d}}{a}_i{J}^i$ where $\mathfrak{d}$ is fixed, $a_i\in \mathcal{C}$ are deterministic scalars and $J$ is the nilpotent matrix $J(i,j)={\mathbf{1}}_{j=i+1}$, then $L_N$ converges, as $N\to \mathrm{\infty }$, to the law of ${\sum }_{i=0}^{\mathfrak{d}}{a}_i{U}^i$ where $U$ is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when $\mathfrak{d}=1$, also of i.i.d.~entries on the diagonals in $M_N$.