Exponential singularity and rank bounds for random symmetric matrices

Let A be an n× n real symmetric random matrix whose entries above the diagonal are independent, each with largest atom at most β∈ (0,1). The diagonal entries have an arbitrary joint distribution and are independent of the entries above the diagonal. We prove that \[ P[rank(A)≤ n-k]≤ 2e-c_βnk, 1≤ k≤ n, \] where c_β>0 depends only on $β$. This extends Han's corank bound to the full range 1≤ k≤ n. No moment assumptions are needed, and the entries need not be centered or identically distributed. In particular, the case $k=1$ gives an exponential singularity bound under the uniform atom assumption. Our argument also gives a new proof of the exponential singularity bound for symmetric Rademacher matrices obtained by Campos, Jenssen, Michelen, and Sahasrabudhe. For every fixed p∈ (0,1), it yields the exponential singularity bound conjectured by Costello and Vu for the adjacency matrix of the Erdős--Rényi random graph $G(n,p)$, where the case $p=1/2$ was previously proved by Han.