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Sharp singularity rates for symmetric sign matrices
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GAME #239
Sharp singularity rates for symmetric sign matrices
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| 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. |
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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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Let An 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.
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