Relative volume of comparable pairs under semigroup majorization

2024/10/30 by Fabio Deelan Cunden, Jakub Czartowski, Cunden, Fabio Deelan +4
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Functional Equations Stability Results #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.2410.23196

openalex publication_date 2024/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Any semigroup S of stochastic matrices induces a semigroup majorization relation \prec^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^S? We review recent asymptotic (n→∞) results and conjectures in the case of majorization relation (when 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 S is the set of upper-triangular stochastic matrices.

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