Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails

We prove a large deviation principle for the largest eigenvalue of Wigner matrices without Gaussian tails, namely such that the distribution tails $\mathbb{P}(\mathrm{\mid }{X}_{1,1}\mathrm{\mid }>t)$ and $\mathbb{P}(\mathrm{\mid }{X}_{1,2}\mathrm{\mid }>t)$ behave like $e^{-b{t}^{\alpha }}$ and $e^{-a{t}^{\alpha }}$ respectively for some $a,b\in (0,+\infty)$and$\alpha\in (0,2)$\mathrm{.}Thelargedeviationprincipleisofspeed$N^{\alpha/2}$ and with a good rate function depending only on the tail distribution of the entries.