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We consider the spectrum of additive, polynomially vanishing randomperturbations of deterministic matrices, as follows. Let $M_N$ be adeterministic $N\times N$ matrix, and let $G_N$ be a complex Ginibre matrix. Weconsider the matrix $\mathcal{M}_N=M_N+N^{-\gamma}G_N$, where $\gamma>1/2$.With $L_N$ the empirical measure of eigenvalues of $\mathcal{M}_N$, we providea general deterministic equivalence theorem that ties $L_N$ to the singularvalues 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$J(i,j)={\bf 1}_{j=i+1}$, then $L_N$ converges, as $N\to\infty$, to the law of$\sum_{i=0}^{\mathfrak{d}} a_i U^i$ where $U$ is a uniform random variable onthe unit circle in the complex plane. We also consider the case of slowlyvarying diagonals (twisted Toeplitz matrices), and, when $\mathfrak{d}=1$, alsoof i.i.d.~entries on the diagonals in $M_N$.

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