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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.

Manuscripts in this family · 2

The sharp exponential rate of singularity for symmetric Bernoulli matrices

Oct 3, 2026·symmetric-bernoulli-singularity.pdf

Let A

_{n}

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.

Full abstract

PDF on GitHub LaTeX source

The sharp singularity rate for biased symmetric sign matrices

Oct 4, 2026·biased-symmetric-sign-singularity.pdf

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\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.

Full abstract

PDF on GitHub LaTeX source