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The sharp singularity rate for biased symmetric sign matrices

OpenAI · October 4, 2026

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Abstract

Let An be a symmetric random matrix whose entries on and above the diagonal are independent signs, equal to 1 with fixed probability $p\in(0,1)\setminus\{1/2\}$. We prove that $\mathbb P(\det A_n=0)=(p^2+(1-p)^2+o(1))^n$. The rate is attained by the event that two rows agree.

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Possibly relevant literature (found by the agent, not checked)

  • Z. Song, Singularity of biased discrete random matrix (arXiv:2509.06408, v1 Sep 2025, v2 Dec 2025). For independent entries with heavily skewed discrete laws, it identifies zero columns and linearly dependent column pairs as the main causes of singularity, the independent-entry counterpart of the claim that agreeing rows attain the biased symmetric rate.
  • V. Jain, A. Sah, M. Sawhney, Sharp invertibility of random Bernoulli matrices (arXiv:2010.06553, 2020). It gives the exact singularity probability of i.i.d. Bernoulli($p$) matrices, confirming a conjecture of Litvak and Tikhomirov, and is a sibling of the cited Jain–Sah–Sawhney "Singularity of discrete random matrices", whose entropy-versus-collision strategy the paper adapts.

Digests and expositions

  • Latent Seminar, Sharp singularity rates for symmetric random sign matrices (Latent Seminar, 2026-10-06). Explains both manuscripts’ equal-row lower bounds, corank-one and completing-column reductions, principal-minor paths, and additional entropy estimates for biased signs, while explicitly disclaiming independent verification of the complete proofs. [family-level]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026,
  author = {{OpenAI}},
  title = {{The sharp singularity rate for biased symmetric sign matrices}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026/biased-symmetric-sign-singularity.pdf}{OAI:The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026}},
  year = {2026}
}

Family 239: Sharp singularity rates for symmetric random sign matrices

Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give $\Pr(\det A_n=0)=(1/2+o(1))^n$; for fixed bias $p\in(0,1)\setminus\{1/2\}$, the rate is $(p^2+(1-p)^2+o(1))^n$. In the biased case, agreeing rows attain this rate.

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