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PDE and mathematical physics · Oct 6, 2026

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⁡An=0)=(1/2+o(1))n\Pr(\det A_n=0)=(1/2+o(1))^n; for fixed bias p∈(0,1)∖{1/2}p\in(0,1)\setminus\{1/2\}, the rate is (p2+(1−p)2+o(1))n(p^2+(1-p)^2+o(1))^n. In the biased case, agreeing rows attain this rate.

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