  We consider $n\times n$ Hermitian matrices with i.i.d. entries $X_{ij}$ whose
tail probabilities $\mathbb {P}(|X_{ij}|\geq t)$ behave like $e^{-at^{\alpha}}$
for some $a>0$ and $\alpha \in(0,2)$. We establish a large deviation principle
for the empirical spectral measure of $X/\sqrt{n}$ with speed $n^{1+\alpha /2}$
with a good rate function $J(\mu)$ that is finite only if $\mu$ is of the form
$\mu=\mu_{\mathrm{sc}}\boxplus\nu$ for some probability measure $\nu$ on
$\mathbb {R}$, where $\boxplus$ denotes the free convolution and
$\mu_{\mathrm{sc}}$ is Wigner's semicircle law. We obtain explicit expressions
for $J(\mu_{\mathrm{sc}}\boxplus\nu)$ 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.
