We study the spectra of general \(N\times N\) Toeplitz matrices given by symbols in the Wiener Algebra perturbed by small complex Gaussian random matrices, in the regime \(N\gg 1\). We prove an asymptotic formula for the number of eigenvalues of the perturbed matrix in smooth domains. We show that these eigenvalues follow a Weyl law with probability sub-exponentially close to 1, as \(N\gg 1\), in particular that most eigenvalues of the perturbed Toeplitz matrix are close to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.

中文翻译：

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服从高斯扰动的一般Toeplitz矩阵

我们研究在系统（\ N \ gg 1 \）中由小复杂高斯随机矩阵扰动的维纳代数中的符号给出的一般\（N \ N N） Toeplitz矩阵的谱。我们证明了光滑域中被摄动矩阵的特征值数量的渐近公式。我们证明这些特征值遵循Weyl定律，其概率次指数接近1，如\（N \ gg 1 \），特别是扰动的Toeplitz矩阵的大多数特征值都接近于由不变的Toeplitz矩阵的符号。