The rank of sparse symmetric matrices over arbitrary fields

Let (Formula presented.) be an arbitrary field and (Formula presented.) be a sequence of sparse weighted Erdős–Rényi random graphs on (Formula presented.) vertices with edge probability (Formula presented.), where weights from (Formula presented.) are assigned to the edges according to a matrix (Formula presented.). We show that the normalized rank of the adjacency matrix of (Formula presented.) converges to a constant, and derive the limiting expression. Our result shows that for the general class of sparse symmetric matrices under consideration, the asymptotics of the normalized rank are independent of the edge weights and even the field, in the sense that the limiting constant for the general case coincides with the one previously established for adjacency matrices of sparse nonweighted Erdős–Rényi matrices over (Formula presented.). Our proof, which is purely combinatorial in its nature, is based on an intricate extension of a novel perturbation approach to the symmetric setting.