Author

Caden Young

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Preprint Oct 2026

Sharp Asymptotics for the Solvability Probability of Random Stable Roommates

For even $n$, let $P_n$ be the probability that independent uniform strict preference lists on $n$ participants admit a stable perfect matching. We prove \[ P_n\sim\frac{e\,2^{1/4}\Gamma(3/4)}{\sqrt\pi}\,n^{-1/4}. \] This establishes Mertens's conjectured exponent of decay, with a leading constant different from his or...

Caden Young, Ian Studebaker · 0 citations

Preprint Jul 2026

Sharp Exponential Asymptotics for Normal Sign Matrices

Let $M_n$ be an $n\times n$ random matrix whose entries are independent Rademacher random variables, and put $N=\binom n2$. We prove \[ Pr(M_nM_n^T=M_n^TM_n)=2^{-N+O(n)}. \] This gives the sharp exponential order for the probability that a random sign matrix is normal. The lower bound is supplied by symmetric sign matr...

Caden Young · 0 citations