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Regular version of the site

Article

Noise sensitivity of the top eigenvector of a Wigner matrix

Probability Theory and Related Fields. 2020. Vol. 177. P. 1103-1135.
Bordenave C., Lugosi G., Zhivotovskiy N.

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×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]. We prove that, with high probability, when 𝑘≪𝑁5/3−𝑜(1), then v and 𝑣[𝑘] are almost collinear and when 𝑘≫𝑁5/3, then 𝑣[𝑘] is almost orthogonal to v.