We study the eigenvalues of the Toeplitz quantization of complex-valued functions on the torus subject to small random perturbations given by a complex-valued random matrix whose entries are independent copies of a random variable with mean 0, variance 1 and bounded fourth moment. We prove that the eigenvalues of the perturbed operator satisfy a Weyl law with probability close to one, which proves in particular a conjecture by Christiansen and Zworski (Commun Math Phys 299:305–334, 2010).

中文翻译：

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几乎可以肯定的量化环面外尔定律

我们研究了复值函数在环面上的托普利茨量化的特征值，该特征值受到由复值随机矩阵给出的小随机扰动的影响，该矩阵的条目是具有均值 0、方差 1 和有界四阶矩的随机变量的独立副本。我们证明扰动算子的特征值满足 Weyl 定律且概率接近 1，这尤其证明了 Christiansen 和 Zworski 的猜想（Commun Math Phys 299:305–334, 2010）。