Sharp singularity asymptotics for iid Bernoulli matrices

Prove that the singularity probability of an n by n iid Bernoulli matrix B_n satisfies P(det(B_n)=0)=(1+o(1))2n^2 2^{-n}.

Background

For iid matrices with entries uniformly distributed in {-1,1}, the dominant known mechanism for singularity is that two rows or columns coincide up to sign, yielding the lower-bound scale 2n2 2{-n}. The paper states that the matching asymptotic upper bound remains unresolved and identifies it as a central open problem.

References

Indeed, it is widely believed that \begin{equation} \label{eq:asym-conj} P(\det(B_n) = 0) = (1+o(1))2n2 2{-n}. \end{equation}

— Probabilistic combinatorics at exponentially small scales  (2512.15077 - Sahasrabudhe, 17 Dec 2025) in Section 4, subsection “Random matrix theory at exponentially small scales”

the conjectured finer asymptotic \begin{equation}\label{eq:square-singularity-conjecture} \mathbb P{M_n\text{ is singular}} =(1+o(1))\,\frac{n2}{2{n-1}} \end{equation} still remains open.

eq:square-singularity-conjecture:

P{Mn is singular}=(1+o(1)) n22n−1\mathbb P\{M_n\text{ is singular}\} =(1+o(1))\,\frac{n^2}{2^{n-1}}

— Boolean threshold functions, neuron capacity, and memory retrieval  (2609.29756 - Xie, 24 Sep 2026) in Section 1.6, “The linear dependence of Rademacher random vectors”