  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 $\gamma>1/2$.
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
$J(i,j)={\bf 1}_{j=i+1}$, 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 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 $M_N$.
