Result 239, Probability and statistical mechanics

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⁡An=0)=(1/2+o(1))n\Pr(\det A_n=0)=(1/2+o(1))^n; for fixed bias p∈(0,1)∖{1/2}p\in(0,1)\setminus\{1/2\}, the rate is (p2+(1−p)2+o(1))n(p^2+(1-p)^2+o(1))^n. In the biased case, agreeing rows attain this rate.

New or sharp bound

The bigger picture

Why it matters

A symmetric matrix filled with random plus and minus signs can still fail to be invertible. These manuscripts quantify how quickly that failure becomes rare as the matrix grows, including when one sign is favored.

What changes?

The model is an n by n symmetric matrix whose entries on and above the diagonal are independent signs; entries below mirror those above. The manuscripts report singularity probabilities of (1/2 + o(1)) to the nth power for uniform signs, and (p squared + (1-p) squared + o(1)) to the nth power when +1 has fixed probability p. The biased result assumes p strictly between zero and one, excluding one half. Each o(1) vanishes as n grows, with p fixed.

What does that help mathematicians do?

In the biased case, the simple event that two rows agree already attains the reported exponential rate. This identifies a concrete obstruction to invertibility that matches the overall probability at that scale: more complicated dependencies cannot produce a larger exponential rate. Researchers can also deduce that any fixed nonuniform bias makes singularity exponentially more likely than uniform signs. These statements do not determine exact finite-size probabilities or establish that agreeing rows explain every failure.

Are there practical applications?

The immediate value is foundational: the results sharpen our understanding of exact linear dependence in random matrices constrained by symmetry. For linear systems with these random coefficient matrices, they quantify the asymptotic chance that a unique solution for every right-hand side is impossible. They do not establish numerical stability or practical finite-size error estimates.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The sharp exponential rate of singularity for symmetric Bernoulli matrices

October 3, 2026 67 pages

Let An be a symmetric n×nn\times n matrix whose entries on and above the diagonal are independent uniform signs. We prove

Pr⁡(det⁡An=0)=(12+o(1))n.\displaystyle \Pr(\det A_n=0)=\left(\frac12+o(1)\right)^n.

PDF LaTeX source
Cite (BibTeX) 
@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

October 4, 2026 34 pages

Let An be a symmetric random matrix whose entries on and above the diagonal are independent signs, equal to 1 with fixed probability p∈(0,1)∖{1/2}p\in(0,1)\setminus\{1/2\}. We prove that P(det⁡An=0)=(p2+(1−p)2+o(1))n\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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.