Precise Delocalisation and Gumbel Laws for Eigenvectors of Wigner Matrices

We prove a delocalisation bound for eigenvectors of Wigner matrices with the precise relationship between the size of the largest entry and the decay exponent of the probability. We also prove that the largest entry of an individual eigenvector and the largest entry of all eigenvectors are both Gumbel distributed. The proof is based on the inclusion-exclusion principle, the small probability comparison of Erdős--Xu, and partial diagonalisation of Gaussian divisible matrices.