The sharp exponential singularity rate of symmetric random sign matrices: P(det A_n = 0) = (1/2 + o(1))^n

Principal theorem: for symmetric $A_n$ with independent uniform signs on and above the diagonal, $\Pr(\det A_n=0)=(1/2+o(1))^n=2^{-n+o(n)}$, via a Schur-complement reduction to symmetric 0/1 matrices, a corank-one reduction and a counting argument for admissible appended columns. Companion: for fixed bias $p\in(0,1)\setminus\{1/2\}$, the rate is $p^2+(1-p)^2$, attained by two agreeing rows. Not shown: the believed $\Theta(n^22^{-n})$ asymptotic, uniformity in $p$, or general entry distributions.

Principal theorem: for symmetric $A_n$ with independent uniform signs on and above the diagonal, $\Pr(\det A_n=0)=(1/2+o(1))^n=2^{-n+o(n)}$, via a Schur-complement reduction to symmetric 0/1 matrices, a corank-one reduction and a counting argument for admissible appended columns. Companion: for fixed bias $p\in(0,1)\setminus\{1/2\}$, the rate is $p^2+(1-p)^2$, attained by two agreeing rows. Not shown: the believed $\Theta(n^22^{-n})$ asymptotic, uniformity in $p$, or general entry distributions.