Non universality of fluctuations of outlier eigenvectors for block diagonal deformations of Wigner matrices

In this paper, we investigate the fluctuations of a unit eigenvector associated to an outlier in the spectrum of a spiked N × N complex Deformed Wigner matrices M N. M N is defined as follows: M N = W N / √ N + A N where W N is an N × N Hermitian Wigner matrix whose entries have a symmetric law µ satisfying a Poincaré inequality and the matrix A N is a block diagonal Hermitian deterministic matrix A N = diag(θ, A N −1), θ ∈ R has multiplicity one and generates an outlier in the spectrum of M N. In the diagonal case, we prove that the fluctuations of a unit eigenvector corresponding to this outlier of M N are not universal: the limiting distribution is the convolution of µ and a Gaussian distribution. In the block diagonal case, the limiting law is replaced by a sequence of approximations with a Gaussian distribution depending on the size N , whose variance may not converge