Rank deficiency of random matrices

TL;DR

This paper investigates the probability that a large random Bernoulli matrix has a corank of at least k, revealing an exponential decay rate in the matrix size.

Contribution

It establishes the asymptotic probability decay rate for the corank of large random Bernoulli matrices, a novel result in random matrix theory.

Findings

Probability that corank ≥ k decays exponentially with matrix size

Explicit asymptotic decay rate of -k for the probability

Provides insight into the rank deficiency behavior of Bernoulli matrices

Abstract

Let $M_n$ be a random $n\times n$ matrix with i.i.d. $\text{Bernoulli}(1/2)$ entries. We show that for fixed $k\ge 1$, \[\lim_{n\to \infty}\frac{1}{n}\log_2\mathbb{P}[\text{corank }M_n\ge k] = -k.\]