Related papers: Relative volume of comparable pairs under semigroup majorization

Relative volume of comparable pairs under semigroup majorization

URL: http://arxiv.org/abs/2410.23196v1
Date: Wed, 30 Oct 2024 16:48:59 GMT
Title: Relative volume of comparable pairs under semigroup majorization
Authors: Fabio Deelan Cunden, Jakub Czartowski, Giovanni Gramegna, A. de Oliveira Junior,
Abstract summary: We review recent results and conjectures in the case of emphmajorization relation. We prove new exact finite-$n$ results in the case of emphUT-majorization relation.
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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 \emph{majorization} relation (when $\mathcal{S}$ is the set of bistochastic matrices), discuss natural generalisations, and prove new exact finite-$n$ results in the case of \emph{UT-majorization} relation, i.e.,\ when $\mathcal{S}$ is the set of upper-triangular stochastic matrices.

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