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 P(|X1,1| > t) and P(|X1,2| > t) behave like e−bt α and e−atα respectively for some a, b ∈ (0,+∞) and α ∈ (0, 2). The large deviation principle is of speed Nα/2 and with an explicit good rate function depending only on the tail distribution of the entries.