Relative volume of comparable pairs under semigroup majorization

• URL: http://arxiv.org/abs/2410.23196v3
• Date: Tue, 17 Dec 2024 10:50:28 GMT
• Title: Relative volume of comparable pairs under semigroup majorization
• Authors: Fabio Deelan Cunden, Jakub Czartowski, Giovanni Gramegna, A. de Oliveira Junior,
• Abstract summary: Any semigroup $mathcalS$ of matrices induces a semigroup majorization relation $precmathcalS$ on the set $Delta_n-1$ of probability $n$-vectors. Pick $X,Y$ at random in $Delta_n-1$: what is the probability that $X$ and $Y$ are comparable under $precmathcalS$?
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• Abstract: Any semigroup $\mathcal{S}$ of stochastic matrices induces a semigroup majorization relation $\prec^{\mathcal{S}}$ on the set $\Delta_{n-1}$ of probability $n$-vectors. Pick $X,Y$ at random in $\Delta_{n-1}$: what is the probability that $X$ and $Y$ are comparable under $\prec^{\mathcal{S}}$? We review recent asymptotic ($n\to\infty$) results and conjectures in the case of majorization relation (when $\mathcal{S}$ is the set of doubly stochastic matrices), discuss natural generalisations, and prove a new asymptotic result in the case of majorization, and new exact finite-$n$ formulae in the case of UT-majorization relation, i.e. when $\mathcal{S}$ is the set of upper-triangular stochastic matrices.

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