Eigenvector statistics of Lévy matrices

We analyze statistics for eigenvector entries of heavy-tailed random symmetric matrices (also called Lévy matrices) whose associated eigenvalues are sufficiently small. show that the limiting law any such entry is non-Gaussian, given by product a normal distribution with another variable depends on location corresponding eigenvalue. Although latter typically nonexplicit, median it inverse one-sided stable law. Moreover, we different same asymptotically independent, but there nontrivial correlations between eigenvectors nearby eigenvalues. Our findings contrast sharply known behavior Wigner and sparse graphs.