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

中文翻译：

---

具有小随机扰动的 Toeplitz 带矩阵

我们研究了 $N\times N$ Toeplitz 带矩阵在 $N\gg 1$ 状态下被小型复杂高斯随机矩阵扰动的光谱。我们证明了概率 Weyl 定律，它为某些域中的特征值数量提供了精确的渐近公式，这可能取决于 $N$，概率次指数（以 $N$ 为单位）接近 $1$。我们表明，对于所有 $\varepsilon >0$，扰动 Toeplitz 矩阵的大多数特征值与曲线中的曲线的距离至多为 $\mathcal{O}(N^{-1+\varepsilon})$由未扰动 Toeplitz 矩阵的符号给出的复平面。