On the rank of a random symmetric matrix in the large‐deviation regime

Abstract Let be an random symmetric matrix with independent identically distributed subgaussian entries of unit variance. We prove the following large‐deviation inequality for the rank of : for all , for some fixed constants . A similar large‐deviation inequality is proven for the rank of the adjacency matrix of dense Erdös–Rényi graphs. This corank estimate enhances the recent breakthrough of Campos, Jensen, Michelen, and Sahasrabudhe that the singularity probability of a random symmetric matrix is exponentially small, and echoes a large‐deviation inequality of Mark Rudelson for the rank of a random matrix with independent entries.