Comparability in Bruhat orders

We determine the sharp asymptotic scale of the probability that two uniformly random permutations are comparable in weak Bruhat order, showing that $\mathbb{P}(σ_1 \preceq_W σ_2)=\exp\Bigl(\bigl(-\tfrac12+o(1)\bigr),n\log n\Bigr)$. 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 $\exp(-Θ(n))$. 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.

💡 Research Summary

The paper investigates the probability that two independent uniformly random permutations of size n are comparable under the weak and strong Bruhat orders on the symmetric group Sₙ. These two partial orders arise naturally in algebraic geometry (flag varieties), representation theory, and combinatorial geometry (the permutohedron). While the comparability probability is a basic global statistic for any finite poset, exact formulas are known only for a few classical lattices; for Bruhat orders, prior work by Hammett and Pittel (2008) gave very coarse bounds: for the weak order, between exp((-1+o(1)) n log n) and exp(-Θ(n)), and for the strong order, between exp(-c n) and 1/n².

Main Results

1. Weak Bruhat Order (Theorem 1.1).
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