Index statistical properties of sparse random graphs

Using the replica method, we develop an analytical approach to compute the characteristic function for the probability $\mathcal{P}_N(K,\lambda)$ that a large $N \times N$ adjacency matrix of sparse random graphs has $K$ eigenvalues below a threshold $\lambda$. The method allows to determine, in principle, all moments of $\mathcal{P}_N(K,\lambda)$, from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with $N \gg 1$ for $|\lambda| > 0$, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erd\"os-R\'enyi and regular random graphs, both exhibiting a prefactor with a non-monotonic behavior as a function of $\lambda$. These results contrast with rotationally invariant random matrices, where the index variance scales only as $\ln N$, with an universal prefactor that is independent of $\lambda$. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.