Relative volume of comparable pairs under semigroup majorization

TL;DR

The paper analyzes when two random probability vectors on the simplex Δ_{n-1} are comparable under semigroup majorization, focusing on both standard majorization and UT-majorization. It provides exact finite-n results for UT-comparability (probability 1/n) and an explicit, simple distribution for the maximal nondeterministic conversion probability Π_{UT}, while delivering a concentration result showing Π(X,Y) → 1 in probability for standard majorization. Central to the analysis are monotone-function frames that yield explicit expressions for conversion probabilities and probabilistic techniques (Sparre–Andersen, Bernstein bounds) alongside Bolshev’s recursion and Dirichlet stick-breaking representations. The findings illuminate the typical convertibility of random states in high dimensions and connect resource-theoretic concepts to concrete geometric and probabilistic structures on the simplex.

Abstract

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.

Figures (3)

• Figure 1: Antecedent sets $\gamma(y, \prec)$ (in green) and $\gamma(y, \prec^{\mathrm{UT}})$ (in red) for each $y \in \Delta_2$ (depicted as a black star), where $y$ is (a) $y = (0.33, 0.18, 0.49)$, (b) $y = (0.60, 0.26, 0.14)$, (c) $y = (0.43, 0.08, 0.49)$, (d) $y = (0.14, 0.31, 0.55)$, (e) $y = (0.42, 0.49, 0.09)$, and (f) $y = (0.10, 0.74, 0.16)$. Points obtained by applying extreme operations with respect to majorization and UT-majorization are depicted by green and red circles, respectively. Note that these are not necessarily extreme points of the corresponding antecedent sets.
• Figure 2: The antecedent set of $y$ is shown for both the weak (top panel) and weak UT (bottom panel) majorizations for (a) $x = \left(\frac{8}{10}, \frac{2}{10}\right)$ (blue circle) and $y = \left(\frac{2}{5}, \frac{3}{5}\right)$ (black star), with $p^{*}x$ (pink circle) lying on the boundary of the antecedent set, and (b) $x = \left(\frac{7}{10}, \frac{1}{4}, \frac{1}{20}\right)$ (blue circle) and $y = \left(\frac{1}{12}, \frac{5}{12}, \frac{1}{2}\right)$ (black star), again with $p^{*}x$ (pink circle) lying on the boundary of the antecedent set.
• Figure 3: Empirical cumulative distribution of (a) $\Pi(X,Y)$ and (b) $\Pi_{\text{UT}}(X,Y)$, obtained from ${10^5}$ samples of uniformly random pairs $(x_i, y_i)$ in the simplex $\Delta_{n-1}$.One can see that the distribution of $\Pi(X,Y)$ concentrates at $1$ as $n\to\infty$, at a slow rate.

Theorems & Definitions (30)

• Theorem A
• Theorem B
• Example 2.1
• Remark 1
• Remark 2: Physics
• Remark 3: Combinatorics
• Theorem 3.1: Cunden2021
• Theorem 3.2: Jain24
• Theorem 3.3
• Proposition 4.1