  An elliptic random matrix $X$ is a square matrix whose $(i,j)$-entry $X_{ij}$
is independent of the rest of the entries except possibly $X_{ji}$. Elliptic
random matrices generalize Wigner matrices and non-Hermitian random matrices
with independent entries. When the entries of an elliptic random matrix have
mean zero and unit variance, the empirical spectral distribution is known to
converge to the uniform distribution on the interior of an ellipse determined
by the covariance of the mirrored entries.
  We consider elliptic random matrices whose entries fail to have two finite
moments. Our main result shows that when the entries of an elliptic random
matrix are in the domain of attraction of an $\alpha$-stable random variable,
for $0<\alpha<2$, the empirical spectral measure converges, in probability, to
a deterministic limit. This generalizes a result of Bordenave, Caputo, and
Chafa\"i for heavy-tailed matrices with independent and identically distributed
entries. The key elements of the proof are (i) a general bound on the least
singular value of elliptic random matrices under no moment assumptions; and
(ii) the convergence, in an appropriate sense, of the matrices to a random
operator on the Poisson Weighted Infinite Tree.
