A large deviation principle for Wigner matrices without Gaussian tails

We consider $n\times n$ Hermitian matrices with i.i.d. entries $X_{ij}$ whose tail probabilities $\mathbb{P}(\mathrm{\mid }{X}_{ij}\mathrm{\mid }\ge t)$ behave like $e^{-a{t}^{\alpha }}$ for some $a>0$ and $\alpha \in (0,2)$. We establish a large deviation principle for the empirical spectral measure of $X\mathrm{/}\sqrt{n}$ with speed $n^{1+\alpha \mathrm{/}2}$ with a good rate function $J(\mu )$ that is finite only if $\mu$ is of the form $\mu ={\mu }_{\mathrm{s}\mathrm{c}}⊞\nu$ for some probability measure $\nu$ on $\mathbb{R}$, where $⊞$ denotes the free convolution and ${\mu }_{\mathrm{s}\mathrm{c}}$ is Wigner’s semicircle law. We obtain explicit expressions for $J({\mu }_{\mathrm{s}\mathrm{c}}⊞\nu )$ in terms of the $\alpha$th moment of $\nu$. The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.