Sharp Singularity Asymptotics for Sparse Bernoulli Matrices

Definitions and mathematical statement

For each integer $n\geq 2$, let $p_n\in[0,1]$, and let $M_n=(M_{ij})_{1\leq i,j\leq n}$ have mutually independent entries with \[ \mathbb P(M_{ij}=1)=p_n,\qquad \mathbb P(M_{ij}=0)=1-p_n. \] Write $R_i$ and $C_j$ for its rows and columns, and define \[ \begin{aligned} E_{\mathrm{zero},n} &=\{\exists i:R_i=0\}\cup\{\exists j:C_j=0\},\\ E_{\mathrm{equal},n} &=\{\exists i<j:R_i=R_j\}\cup\{\exists i<j:C_i=C_j\}. \end{aligned} \] Under the usual characteristic-zero interpretation, define the singularity event by \[ S_n=\{\det M_n=0\}, \] where the determinant is evaluated over ${\mathbb{R}}$, equivalently over ${\mathbb{Q}}$. The source does not explicitly specify the field, so this interpretation requires confirmation; singularity over a finite field would be a different question.

The original supplied formulation specifies the sparse regime $p_n\to0$. The conjectural target is to determine the range of such sequences for which \[ \mathbb P(S_n) =(1+o_n(1))\mathbb P\left(E_{\mathrm{zero},n}\cup E_{\mathrm{equal},n}\right) \qquad(n\to\infty). \] Here $o_n(1)$ denotes a quantity tending to zero as $n\to\infty$; no uniformity over parameter sequences is specified.

The supplied corrected statement additionally asks whether, when $\limsup_{n\to\infty}p_n<\tfrac12$, \[ \mathbb P(S_n)=2n(1+o_n(1))(1-p_n)^n. \] Unresolved qualification. The condition $\limsup p_n<\tfrac12$, even together with $p_n\to0$, does not supply a sufficient parameter range for this second formula as written. For example, it allows $p_n=n^{-2}$, for which $2n(1-p_n)^n\to\infty$, whereas a probability is at most one. An additional restriction is therefore missing; none is imposed here. It is also unclear whether this second question is intended only within $p_n\to0$ or in a wider range satisfying the displayed upper restriction.

The original source summary reports a bound \[ \mathbb P(S_n)\leq\exp(-c n p_n) \qquad\text{for }p_n\geq C\frac{\log n}{n} \] in its sparse-regime discussion, and asks for improvement to the conjectured exponent. The constants $c,C>0$, their quantifiers, any further parameter restrictions, and the precise meaning of that exponent improvement are not supplied. This bound is retained as reported background, not asserted here as an independently verified theorem.

Short English statement

For independent Bernoulli matrices in the sparse regime $p_n\to0$, ask when singularity is asymptotically accounted for by a zero row or column, or by two equal rows or columns. The additional proposed asymptotic $2n(1-p_n)^n$ needs a missing restriction on $p_n$. The field and exact parameter ranges require review.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $30004385$, title Sharp Singularity Asymptotics for Sparse Bernoulli Matrices; collection: Oberwolfach Reports Open Problems; source problem code catalogue entry. The proposed year is recorded as 2020; no individual proposer is supplied.
• Supplied report citation: New Perspectives and Computational Challenges in High Dimensions (2021), Oberwolfach Reports, pp. 377--416, DOI 10.4171/owr/2020/6. https://doi.org/10.4171/owr/2020/6. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Litvak and Tikhomirov, https://arxiv.org/abs/2004.03131. No title or theorem locator is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that sharp singularity asymptotics have been proved through an important logarithmic sparse regime, improving earlier bounds, while the conjectured description in remaining sparser regimes is not fully established. Both links above are listed as sources checked. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified. Its verification note reports that the official report was checked and that the two exceptional events and both conjectured asymptotic formulas were restored. This records the supplied verification claim; no source visit or independent verification is asserted here.