Localization and delocalization for heavy tailed band matrices

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 $\alpha \text{textless}2(1+{\mu }^{-1})$, the largest eigenvalues have order $N^{(1+\mu )\mathrm{/}\alpha }$, 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 $\alpha \text{textless}2$, and by Auffinger, Ben Arous and P{'e}ch{'e} when $\alpha \text{textless}4$). On the other hand, when $\alpha \text{textgreater}2(1+{\mu }^{-1})$, the largest eigenvalues have order $N^{\mu \mathrm{/}2}$ and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their $N$ coordinates.