  We consider some random band matrices with band-width $N^\mu$ whose entries
are independent random variables with distribution tail in $x^{-\alpha}$. We
consider the largest eigenvalues and the associated eigenvectors and prove the
following phase transition. On the one hand, when
$\alpha\textless{}2(1+\mu^{-1})$, the largest eigenvalues have order
$N^{(1+\mu)/\alpha}$, are asymptotically distributed as a Poisson process and
their associated eigenvectors are essentially carried by two coordinates (this
phenomenon has already been remarked by Soshnikov for full matrices with heavy
tailed entries,i.e. when $\alpha\textless{}2$, and by Auffinger, Ben Arous and
P{\'e}ch{\'e} when $\alpha\textless{}4$). On the other hand, when
$\alpha\textgreater{}2(1+\mu^{-1})$, the largest eigenvalues have order
$N^{\mu/2}$ and most eigenvectors of the matrix are delocalized, i.e.
approximately uniformly distributed on their $N$ coordinates.
