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We study the spectra of $N\times N$ Toeplitz band matrices perturbed by smallcomplex Gaussian random matrices, in the regime $N\gg 1$. We prove aprobabilistic Weyl law, which provides an precise asymptotic formula for thenumber of eigenvalues in certain domains, which may depend on $N$, withprobability sub-exponentially (in $N$) close to $1$. We show that mosteigenvalues of the perturbed Toeplitz matrix are at a distance of at most$\mathcal{O}(N^{-1+\varepsilon})$, for all $\varepsilon >0$, to the curve inthe complex plane given by the symbol of the unperturbed Toeplitz matrix.

Toeplitz band matrices with small random perturbations