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The sharp exponential rate of singularity for symmetric Bernoulli matrices
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Abstract
Let An 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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Possibly relevant literature (found by the agent, not checked)
- X. Hu, Z. Song, X. Wu, Sharp singularity asymptotics for discrete random matrices (arXiv:2609.39463, v1 30 Sep 2026, v2 16 Oct 2026). For independent (non-symmetric) uniform signs it proves the full asymptotic $(2+o(1))n^2 2^{-n}$, a sharper form of the Tikhomirov rate the introduction compares Theorem 1.1 with; the companion biased paper cites it, but this paper does not.
- A. Ferber, V. Jain, A. Sah, M. Sawhney, Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial (arXiv:2106.04049, June 2021). It proves rank-distribution estimates over $\mathbb F_p$ for symmetric $\pm1$ matrices, which bear on the paper's step that bounds "coranks modulo all primes" along the minor path.
- V. Jain, A. Sah, M. Sawhney, Sharp invertibility of random Bernoulli matrices (arXiv:2010.06553, 2020). It proves a sharp $(1/2+o(1))^n$ singularity rate for a model with dependent entries (rows uniform on the central slice), which bears on the introduction's point that dependence between rows calls for a different source of anti-concentration.
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
- VibeMathed: Problem record
- Emergent Mind: family 239
Independent sites; their content is not reviewed here.
Cite
@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}
}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.
Other papers in this family:
- The sharp singularity rate for biased symmetric sign matrices · October 4, 2026