{"id":"openai-math/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices","pdf":"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","url":"https://github.com/openai/math/tree/main/preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026","tags":["random-matrices","probability"],"score":0,"title":"The sharp singularity rate for biased symmetric sign matrices","domain":"Probability and statistical mechanics","family":"239","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-04","reactions":{},"formalized":false,"family_title":"Sharp singularity rates for symmetric random sign matrices"}