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

TL;DR

This paper establishes large deviation inequalities for the rank of random symmetric matrices with subgaussian entries and for dense Erdős-Rényi graph adjacency matrices, improving understanding of their rank distribution.

Contribution

It proves new large deviation bounds for the rank of symmetric random matrices and dense graph adjacency matrices, extending recent results on singularity probability.

Findings

Probability that the rank is close to full is exponentially high.

Provides bounds for the probability that the rank drops significantly.

Enhances understanding of the rank distribution in large random matrices.

Abstract

Let $A$ be an $n\times n$ random symmetric matrix with independent identically distributed subgaussian entries of unit variance. We prove the following large deviation inequality for the rank of $A$: for all $1\leq k\leq c\sqrt{n}$, $$\mathbb{P}(\operatorname{Rank}(A)\geq n-k)\geq 1-\exp(-c'kn),$$ for some fixed constants $c,c'>0$. A similar large deviation inequality is proven for the rank of the adjacency matrix of dense Erdos-Renyi 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.