Relative volume of comparable pairs under semigroup majorization

TL;DR

This paper investigates the probability that two randomly chosen probability vectors are comparable under semigroup majorization relations, reviewing recent asymptotic results, discussing generalizations, and presenting new asymptotic and exact finite-$n$ results.

Contribution

It introduces new asymptotic results for majorization and provides exact finite-$n$ formulas for UT-majorization, expanding understanding of comparability under semigroup majorization.

Findings

Asymptotic probability results for majorization relations

New exact formulas for UT-majorization at finite $n$

Discussion of generalizations of majorization relations

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.