Probability and statistical mechanics · Result 239

Sharp singularity rates for symmetric random sign matrices

2 papers

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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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

The sharp exponential rate of singularity for symmetric Bernoulli matrices

OpenAI · October 3, 2026

Let A_n be a symmetric \(n\times n\) matrix whose entries on and above the diagonal are independent uniform signs. We prove \(\displaystyle \Pr(\det A_n=0)=\left(\frac12+o(1)\right)^n.\)

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@misc{OAI:The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026,
  author = {{OpenAI}},
  title = {{The sharp exponential rate of singularity for symmetric Bernoulli matrices}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026/symmetric-bernoulli-singularity.pdf}{OAI:The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026}},
  year = {2026}
}

The sharp singularity rate for biased symmetric sign matrices

OpenAI · October 4, 2026

Let A_n 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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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}
}

Formal verification

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