OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
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∈(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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Let A_(n) be a symmetric n× n matrix whose entries on and above the diagonal are independent uniform signs. We prove \displaystyle \Pr(\det A_n=0)=(\frac12+o(1))^n.
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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∈(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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