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.