Singularity of discrete random matrices.
Let $\xi$ be a non-constant real-valued random variable with finite support, and let $M_{n}(\xi)$ denote an $n\times n$ random matrix with entries that are independent copies of $\xi$. We show that, if $\xi$ is not uniform on its support, then
\begin{align*}
\mathbb{P}[M_{n}(\xi)\text{ is singular}] &= (1+o_n(1))\mathbb{P}[\text{zero row or column, or two equal (up to sign) rows or columns}].
\end{align*}
For $\xi$ which is uniform on its support, we show that
\begin{align*}
\mathbb{P}[M_{n}(\xi)\text{ is singular}] &= (1+o_n(1))^{n}\mathbb{P}[\text{two rows or columns are equal}].
\end{align*}
Corresponding estimates on the least singular value are also provided. These results, combined with works of Basak and Rudelson, Huang, and Tikhomirov and Litvak answer the singularity question for $\xi = \text{Ber}(p_n)$ when $\limsup p_n < 1/2$. For fixed $p \in (1/2, 1)$, these results provide the leading term in the asymptotic expansion of the singularity probability of $M_{n}(\text{Ber}(p))$, whereas even the exponential rate was previously unknown. Finally, for fixed $p\in(0,1/2)$, a refined analysis provides the first \emph{two} terms of the asymptotic expansion for the singularity probability.
This is joint work with Vishesh Jain.