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.

Manuscripts (2)

The sharp exponential rate of singularity for symmetric Bernoulli matrices

October 3, 2026 · source · cite as \cite{OAI:The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-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.

The sharp singularity rate for biased symmetric sign matrices

October 4, 2026 · source · cite as \cite{OAI:The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-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.

Result 239 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.