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

Abstract

Let Ξ be the adjacency matrix of an Erdős-Rényi graph on n vertices and with parameter p and consider A a $n\times n$ centred 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 $(A\circ \Xi )/\sqrt{np}$ 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.