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