Comparability in Bruhat orders

TL;DR

This paper precisely determines the asymptotic probability that two random permutations are comparable in weak Bruhat order, significantly improving previous bounds and also providing new lower bounds for strong Bruhat order comparability.

Contribution

It establishes the exact asymptotic scale of comparability probabilities in weak Bruhat order and improves bounds for strong Bruhat order, using advanced combinatorial and probabilistic tools.

Findings

Weak Bruhat order comparability probability is $oxed{ ext{exp}((-1/2+o(1)) n ext{log} n)}$

Improved lower bounds for strong Bruhat order comparability are established

The analysis combines tableau theory, Plancherel measure, and random walk techniques.

Abstract

We determine the sharp asymptotic scale of the probability that two uniformly random permutations are comparable in weak Bruhat order, showing that $\mathbb{P}(\sigma_1 \preceq_W \sigma_2)=\exp\Bigl(\bigl(-\tfrac12+o(1)\bigr)\,n\log n\Bigr)$. This significantly improves both of the best known bounds, due to Hammett and Pittel, which placed this probability between $\exp((-1+o(1))n\log n)$ and $\exp(-\Theta(n))$. We also improve the best known lower bound for strong Bruhat-order comparability, due to the same authors, by proving a subexponential lower bound. The Bruhat orders are natural partial orders on the symmetric group, appearing in wide-reaching settings including the geometry of flag manifolds, the representation theory of $\mathfrak{S}_{n}$, and the combinatorics of the permutohedron. To analyze weak Bruhat order, we combine classic analytic, tableau-theoretic, and poset-theoretic tools, including the Plancherel measure and the RSK bijection. For strong Bruhat order we construct large families where members are comparable with high probability. Our proof that members are comparable combines the tableau criterion with an associated random-walk-type deviation process.