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arXiv · 2602.16625

Comparability of random permutations in the strong Bruhat order

Nicholas Christo·Marcus Michelen
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Abstract

The (strong) Bruhat order for permutations provides a partial ordering defined as follows: two permutations are comparable if one can be obtained from the other by a sequence of adjacent transpositions that each increase the number of inversions by $1$. Given two random permutations, what is the probability that they are comparable in the Bruhat order? This problem was first considered in a 2006 work of Hammett and Pittel, which showed an exponential lower bound and a polynomial upper bound. The lower bound was very recently improved to the subexponential bound of $\exp(-n^{1/2 + o(1)})$ by Boretsky, Cornejo, Hodges, Horn, Lesnevich, and McAllister. Hammett and Pittel predicted that the probability should decrease polynomially. We show that the probability decreases faster than any polynomial and is on the order of $\exp(-\Theta(\log^2 n))$.

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Nicholas Christo, Marcus Michelen. 2026-02-18. Comparability of random permutations in the strong Bruhat order. https://arxiv.org/abs/2602.16625

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