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

We prove a large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails, that is, the distribution tails of the diagonal entries P ( |X1,1|>t) and off-diagonal entries P (|X1,2|>t) behave like e^-btα and e^-atα respectively, for some a,b∈ (0,+∞ ) and α ∈ (0,2). The large deviations principle is of speed Nα /2, and with a good rate function depending only on the distribution tail of the entries.