Relative volume of comparable pairs under semigroup majorization
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915647401754624 |
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| author | Cunden, Fabio Deelan Czartowski, Jakub Gramegna, Giovanni Junior, A. de Oliveira |
| author_facet | Cunden, Fabio Deelan Czartowski, Jakub Gramegna, Giovanni Junior, A. de Oliveira |
| contents | Any semigroup $\mathcal{S}$ of stochastic matrices induces a semigroup majorization relation $\prec^{\mathcal{S}}$ on the set $Δ_{n-1}$ of probability $n$-vectors. Pick $X,Y$ at random in $Δ_{n-1}$: 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_23196 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative volume of comparable pairs under semigroup majorization Cunden, Fabio Deelan Czartowski, Jakub Gramegna, Giovanni Junior, A. de Oliveira Mathematical Physics Combinatorics Probability Quantum Physics Any semigroup $\mathcal{S}$ of stochastic matrices induces a semigroup majorization relation $\prec^{\mathcal{S}}$ on the set $Δ_{n-1}$ of probability $n$-vectors. Pick $X,Y$ at random in $Δ_{n-1}$: 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. |
| title | Relative volume of comparable pairs under semigroup majorization |
| topic | Mathematical Physics Combinatorics Probability Quantum Physics |
| url | https://arxiv.org/abs/2410.23196 |