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.
Categories: math.PR