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Localization and delocalization for heavy tailed band matrices

We consider some random band matrices with band-width $N^μ$ whose entries are independent random variables with distribution tail in $x^{-α}$. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $α\textless{}2(1+μ^{-1})$, the largest eigenvalues have order $N^{(1+μ)/α}$, 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 $α\textless{}2$, and by Auffinger, Ben Arous and P{é}ch{é} when $α\textless{}4$). On the other hand, when $α\textgreater{}2(1+μ^{-1})$, the largest eigenvalues have order $N^{μ/2}$ and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their $N$ coordinates.