On random matrices with large corank
Abstract
Abstract Let $1\le k\le n$ and $M$ be a random $n\times n$ matrix with independent uniformly random $\{\pm 1\}$-entries. We show that there exists an absolute constant $c> 0$ such that $$ \begin{align*} &\mathbf{P}[\operatorname{rank}(M)\le n-k]\le \exp(-c nk).\end{align*} $$ This confirms a well-known prediction in the area, extending a result of Rudelson (who previously proved this same result under the restriction $k\le \sqrt n$, via different methods).
Fields of Study & Subjects
Random Matrices and Applications
advanced mathematical theories
Publication details
- Journal / source
- International Mathematics Research Notices
- Volume
- 2026
- Issue
- 12
- Publication date
- 1 June 2026
- Language
- EN