Litvak–Tikhomirov sparse Bernoulli singularity conjecture

Let $B_n(p_n)$ be an $n\times n$ random matrix with independent and identically distributed Bernoulli entries of mean $p_n$, and let $q_n(p_n)=\mathbb{P}[B_n(p_n)\text{ is singular}]$. Suppose that $(p_n)$ is a sequence of real numbers satisfying

$$0<\liminf p_n\leq\limsup p_n<1/2.$$

Litvak–Tikhomirov conjecture. Then

$$q_n(p_n)=(1+o_n(1))\mathbb{P}[B_n(p_n)\emph{ has a zero row or column}]=(2+o_n(1))n(1-p_n)^n.$$

This is a sparse-regime refinement in which zero rows and columns are exponentially more likely than pairs of equal rows or columns. The source states that the paper’s theorem resolves this conjecture.