Large deviations of the empirical spectral measure of supercritical sparse Wigner matrices

Let $\mathrm{\Xi }$ be the adjacency matrix of an Erdős-Rényi graph on $n$ vertices and with parameter $p$ and consider $A$ a $n\times n$ centered random symmetric matrix with bounded i.i.d. entries above the diagonal. When the mean degree $np$ diverges, the empirical spectral measure of the normalized Hadamard product $(A\circ \mathrm{\Xi })\mathrm{/}\sqrt{np}$ converges weakly in probability to the semicircle law. In the regime where $p\ll 1$ and $np\gg \log n$, we prove a large deviations principle for the empirical spectral measure with speed $n^{2}p$ and with a good rate function solution of a certain variational problem. The rate function reveals in particular that the only possible deviations at the exponential scale $n^{2}p$ are around measures coming from Quadratic Vector Equations. As a byproduct, we obtain a large deviations principle for the empirical spectral measure of supercritical Erdős-Rényi graphs.