We investigate the noise sensitivity of the top eigenvector of a Wigner matrix in the following sense. Let v be the top eigenvector of an $$N\times N$$ N × N Wigner matrix. Suppose that k randomly chosen entries of the matrix are resampled, resulting in another realization of the Wigner matrix with top eigenvector $$v^{[k]}$$ v [ k ] . We prove that, with high probability, when $$k \ll N^{5/3-o(1)}$$ k ≪ N 5 / 3 - o ( 1 ) , then v and $$v^{[k]}$$ v [ k ] are almost collinear and when $$k\gg N^{5/3}$$ k ≫ N 5 / 3 , then $$v^{[k]}$$ v [ k ] is almost orthogonal to v .

中文翻译：

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维格纳矩阵顶部特征向量的噪声灵敏度

我们在以下意义上研究了 Wigner 矩阵的顶部特征向量的噪声敏感性。设 v 是 $$N\times N$$ N × N Wigner 矩阵的顶部特征向量。假设对矩阵的 k 个随机选择的条目进行重新采样，从而产生具有顶部特征向量 $$v^{[k]}$$ v [ k ] 的 Wigner 矩阵的另一个实现。我们证明，当 $$k \ll N^{5/3-o(1)}$$ k ≪ N 5 / 3 - o ( 1 ) ，那么 v 和 $$v^{[k ]}$$ v [ k ] 几乎共线，当 $$k\gg N^{5/3}$$ k ≫ N 5 / 3 ，则 $$v^{[k]}$$ v [ k ]几乎与 v 正交。